Analysis of classical and quantum interacting particle systems

Bogusław Zegarliński Department of Mathematics, Imperial College, London, UK

September 2000

Abstract The course includes a description of construction of Markov semigroups and some methods of controlling their ergodicity in the context of classical and quantum interacting particle systems.

List of Content

Introduction Lecture I : Construction and Ergodicity of Jump Type Dynamics Lecture II : Hypercontractivity and Strong Ergodicity for Classical Systems Lecture III : Logarithmic Sobolev Inequality in Non-Commutative Setting Lecture IV: Particle Structure and Hypercontractivity

Introduction

In these lectures we describe a part of development in the theory of dissipative semigroups on noncommutative infinite dimensional spaces which took place over the last decade. One of the key challenging problems in the noncommutative analysis is a construction of dissipative semigroup which are additionally symmetric in a Hilbert space associated to an a’priori given state on noncommutative algebra. Naturally we would also like to develope some tools to study their ergodicity. Although the understanding of the dissipative semigroups in noncommutative situation remains far behind the commutative development, (in many respects we have here still a gap of 30 - 40 years), one of good news is that recently we begin to be able to construct nontrivial translation invariant symmetric semigroups of jump and diffusion types. To have possibly focused exposition, in this lectures to a large extend we concentrate on the dynamics of jump type ; (for existence result of dissipative dynamics of diffusion type based on strong asymptotic abelianes assumption we refer to [27] - [29] and [37]). In Lecture I we present a general construction of a family of dissipative dynamics living on the inductive limit C^{∗} algebra which generators contain a hamiltonian term. The simplest examples generalise the jump type dynamics (of Glauber and Kawasaki) for classical discrete spin systems for which hamiltonian dynamics does not make natural sense. The definition of the corresponding generator involves generalised conditional expectations ; (more examples together with slightly expanded description of generalised conditional expectations is contained in Appendix I). We describe there a strategy for ergodicity essentially based on the idea of Aizenman and Holey, [3]. The key point of this method is that it allows to formulate a strong ergodicity condition for a dynamics involving jumps of large blocks of degrees of freedom. Similarly as in the classical case, using equivalence of Dirichlet forms, one can conclude a weaker ergodicity in L_{2} sense for an extensive family of jump type dynamics. The disadvantage of this strategy is that when one approaches ”a critical point”, the strong ergodicity condition requires the size of the block to grow to infinity. This means that effectively one cannot control the strong ergodicity for any given dissipative dynamics. A new additional interesting feature of the noncommutative setting is that one can make perturbation of a given jump type dynamics associated to some potential by a hamiltonian term associated to a different interaction. In this case we do not know a’priori the invariant states, though, if the coupling constant is small and the size of the blocks used in the definition of jump type generator is sufficiently large, one can use that strategy to get the strong decay to equilibrium to a unique invariant state. It is an interesting question, (which seems to be impossible to solve by this strategy), whether or not the ergodicity still survives for large coupling of the hamiltonian term. For commutative systems a stronger strategy allowing to overcome the above mentioned problem was introduced by Holey and Stroock, [25]. The key element of this new strategy is the hypercontractivity property of the semigroup in the family of L_{p} spaces. (A dissipative semigroup is naturally contractive in each L_{p} space. The hypercontractivity is a stronger property which allows one to go in the opposite direction than in the Hölder inequality.) Similarly as the dissipativity property, the hypercontractivity allows an equivalent infinitesimal condition, [21], called the Logarithmic Sobolev inequality. This condition says that the relative entropy can be bounded in terms of the Dirichlet form of the generator. The essence of its power is contained in the fact that such the entropy bound, if shown for one generator, it also holds for any other generator with equivalent or larger Dirichlet form. In Lecture II we present the basic features of the hypercontractivity theory and briefly review some applications; for a more comprehensive exposition see [23]. As the entropy has an important meaning in the context of statistical mechanics, one can expect that hypercontractivity can be directly related to some physical characteristics of a large interacting system. A result in this direction, establishing equivalence of hypercontractivity and a number of other conditions from statistical mechanics is presented at the end of this lecture. In particular this includes an equivalence of hypercontractivity with fast decay of correlations in the equilibrium state. In Lecture III, following [36], we establish the fundamentals of noncommutative hypercontractivity theory. We begin there from introducing an interpolating family of noncommutative 𝕃_{p} spaces and studying their geometric and analytic features. In particular for our applications we have to establish relations of positivity in algebra and in the 𝕃_{p} spaces. We also need to study the appropriate isometric embeddings and differentiability property of the norms corresponding to a natural flow related to these embedings. This last property leads us to an intrinsic notion of entropy and its corresponding operator valued counterpart. One can see there that noncommutativity results with a new interesting effect of ”renormalisation” which trivialises in abelian situation. Our construction also assures that the noncommutative entropy has appropriate covariance property (required for logarithmic Sobolev inequality). In this framework we recover basically all essential properties of the hypercontractivity theory and the related infinitesimal condition; (the product property being considered in Lecture IV). The implication of the Logarithmic Sobolev inequality from hypercontractivity follows relatively easily from properties of differentiation of the norm and related covariance property of the entropy. On the other hand to establish the sufficiency proof of the integration lemma one needs suitable regularity properties of the corresponding 𝕃_{p}- Dirichlet forms. In commutative setting , (or in case when the state is given by a trace), these regularity properties follow easily from the representation theorem for Dirichlet forms, but are not in general obvious in noncommutative spaces. It turns out also that the tensorisation property of logarithmic Sobolev inequality appears to be difficult to prove directly, though we essentially recover it in the next lecture by a different method.

To relate our framework to other studies on this subject, (see e.g. [22], [13], [10], [8]), we remark that we do not assume that our underlying algebra is generated by some commutation relations. In Lecture IV we show that the hypercontractivity of a semigroup is strongly related to the particle structure of the corresponding system, (that is to the structure of the corresponding spectral theory of the generator). As discussed in Lecture II and III the hypercontractivity implies the spectral gap property. Using Logarithmic Sobolev inequality one can also establish Gaussian type bounds on moments of Lipschitz random variables. In this lecture, (following an extension of the idea of [38] provided by [11] and [24]), we formulate general conditions on the structure of the spectrum and invariant subspaces of the generator of dissipative dynamics which implies hypercontractivity. Besides appropriate behaviour of the spectrum related to the invariant subspaces, one needs to require Gaussian bounds and weak higher order correlations between various invariant subspaces. Later in this lecture we present an application of this strategy to establish hypercontractivity for product semigroups. As shown in [11] such strategy can be applied to a more complicated situation where the state is not given by a product measure; (as for example Glauber dynamics of one dimensional Ising model at any positive temperature). We mention that the particle structure is of high importance in field theory and statistical mechanics where some interesting, though far from complete, results were established (see e.g. [19] and [35], and references therein). One could hope that use of hypercontractivity could possibly help to achieve some further progress in this direction.

The noncommutative content of these lectures is based on a series of works with T. Bodineau, W. Hebisz, A. Majewski and R. Olkiewicz to whom the author is thankful for collaboration.

Lecture I

Construction and Ergodicity of Jump Type Dynamics

Introduction

In this lecture we present an adaptation of classical methods of construction and studying the ergodicity of infinite dimensional stochastic dynamics P_{t}^{X} to certain quantum lattice systems described by an inductive product C^{∗} algebra. We show how one can construct Markov semigroups which are translation invariant, symmetric in an appropriate Hilbert space (associated to a Gibbs state) and possess some natural regularity property (the Feller property). Although the method applies to a more general situation, for simplicity to a large extend we concentrate the exposition on the case when the generator consists of two parts : one corresponding to a classical potential and describing jumps and the other given by a derivation corresponding to a general quantum hamiltonian dynamics. One can say that the considered system dissipates through some degrees of freedom, while all the others are subjected to a reversible quantum evolution. This is qualitatively new situation which is not available in the classical discrete spin systems. In a special situation when the jump and hamiltonian parts commute with each other, the restriction of the stochastic dynamics to the classical subalgebra coincides with the conventional Glauber type dynamics. However on the full algebra, even in this case we get a nontrivial extension, as the ergodicity may not be in general obvious (despite ergodicity of its classical part). In such the example the family of invariant states include the corresponding Gibbs states with the jump part being symmetric in a corresponding appropriately chosen Hilbert space. In general, since the jump and hamiltonian parts of the generator can be chosen independently, we may not know a’priori an invariant state ; (though in case the Feller property is satisfied, we know that the invariant states exist). We show that if the jump part of the generator defines a strongly ergodic semigroup, it also does if the hamiltonian perturbation is small in an appropriate sense.

The organisation of this lecture is as follows. In the first part we introduce the basic framework of the inductive product C^{∗} algebras and describe elementary jump type Markov generators using generalised conditional expectations. We do that by analogy with the classical block-spin flip generators taking care of the requirements the generator should satisfy in the current more general noncommutative setting. (In the Appendix I at the end of these lectures more examples of jump type generators are presented.) Later we present the main results concerning existence and strong ergodicity of infinite volume Markov semigroup. The idea of the proof of exponential decay to equilibrium is similar to the one introduced in the classical context by Aizenman and Holley, [3], (see also [40], [28]), who showed that under some natural and relatively weak conditions on the decay of correlations for conditional expectations associated to large blocks, (guaranteeing the uniqueness of the Gibbs measure corresponding to a given a’priori conditional expectations), in the commutative context the corresponding block-spin flip dynamics are strongly ergodic in the uniform norm. Because the generators associated to various blocs have equivalent quadratic forms, this ergodicity result implies exponential decay to equilibrium in the corresponding L_{2} space. As we show similar equivalence of the quadratic forms is true in the noncommutative context and so similar L_{2} ergodicity follows.

The basic drawback of the strategy of Aizenman and Holley is that in the situation of a spin system in a neigbourhood of a critical point, one can have a sufficiently strong decay of correlations only on a sufficiently large scale which diverges to infinity when the critical point is approached. This implies that such the method is not suitable to control strong ergodicity of the jump type dynamics associated to any block of a given finite size. A strategy how to overcome this problem for classical spin systems is discussed in Lecture II. One of the ingredients necessary for implementing such strategy is a more detailed control of the approximation of an infinite dimensional semigroup by a finite dimensional ones, (for the last the strong technical methods being well known). The details of the finite dimensional approximation are described after the proof of strong ergodicity. Both of them are based on some regularity properties of elementary generators which are described in a later section.

This lecture is based on the following main sources [27] - [30] in which one can find additional information.

Semiclassical Dissipative Dynamics at Finite Temperatures for Quantum Spin System on a Lattice :

Let M be the single (quantum) spin algebra consisting of n×n complex matrices. Let A , A_{X} , and A_{loc} denotes the corresponding inductive limit algebra over M , the subalgebra of elements localized in a finite set X ⊂⊂ ℤ^{d} and the subalgebra of local elements (generated by all A_{Y} , Y ⊂⊂ ℤ^{d} ), respectively. That is, briefly speaking, (see e.g. [9] for details), each A_{X} is isomorphic to M^{⊗X} , one has natural inclusion condition A_{Y} ⊂ A_{X} for Y ⊂ X and the algebra A is furnished with a norm || · ||_{A} (naturally inherited from the finite tensor products algebras) so that A_{loc} is dense in A. (Later on, if no confusion can arise, we will frequently simplify the notation of the norm by omitting its explicit relation to the algebra.) For an unbounded set Λ ⊆ ℤ^{d}, A_{Λ} is defined as the closure of an algebra generated by all A_{Y} , Y ⊂⊂ Λ . We will be adding superscripts + and h to distinguish the corresponding subalgebras of positive and hermitian elements, respectively. Let M_{c} be a single spin space for a classical single spin system, such that #M_{c} = n. Given a representation of M, we identify the set of diagonal elements diag(M) in M with the (complex valued) functions C(M_{c}) on M_{c}. (Such identification depends on the choice of the representation of the matrices and therefore it can be done in many ways.) That is we are given a bijection ι : C(M_{c}) → diag(M). The inductive limit algebra C over diag(M), called later on the classical subalgebra, can be identified with the set C(Ω_{c}) of continuous functions on the space Ω_{c} ≡ M_{c}^{ℤ^{d}} of configurations of a classical spin system. For f ∈ C its unique correspondent in C(Ω_{c}) will be denoted by f_{c}. In this setting for any f ∈ C , we have

Tr f = μ_{0}f_{c}

where Tr is the normalized trace and μ_{0} denotes the free measure on Ω_{c} defined as the product of uniform probability measures on M_{c}.

Let Φ ≡ {Φ_{X} ∈ A_{X}^{h}}_{X⊂⊂ℤ^{d}} be an interaction potential for a spin system such that

||Φ||_{1} ≡ sup_{i∈ℤ^{d}} ∑_{X∋i} ||Φ_{X}||_{A} < ∞ .  (1)

The class of potentials satisfying (1) will be denoted by B_{1} . A potential Φ in B_{1} such that Φ ≡ {Φ_{X} ∈ A_{X}^{h} ∩ C}_{X⊂⊂ℤ^{d}} will be called a semi-classical potential. It corresponds via bijection ι to a unique classical potential defined correspondingly as a collection of functions on the configuration space Ω_{c}. A potential is of finite range R iff Φ_{X} ≡ 0, (and so ι(Φ_{X}) ≡ 0), for all diam(X) > R. Later on, if there will be no danger of confusion, we will denote the classical potential and its image via ι with the same symbol.

Let ω_{Φ} be a Gibbs state on A. It is known, (see e.g. [9]), that it can be represented as follows

ω_{Φ}(f) ≡ F- lim Tr_{Λ}(ρ_{Λ}f)

the notation F-lim means that Λ invades ℤ^{d} via a suitable increasing van-Hove sequence, ρ_{Λ} ∈ A_{Λ} denoting a suitable normalised density matrix with respect to the partial trace Tr_{Λ} , Λ ⊂⊂ ℤ^{d} , which can be defined for any bounded set X ⊂⊂ ℤ^{d} as follows

Tr_{X}(f) ≡ ∫ ν_{X}(dU)U^{∗}fU,  (2)

with ν_{X}(dU) denoting the (normalised) Haar measure on the unitary group of A_{X} . (Such the formula for the partial trace proves to be very convenient when passing to a representation.)

Exercise : Prove that the partial trace Tr_{X} has the following properties of conditional expectation.

(i) Tr_{X}f^{∗}f ≧ 0

(ii) Tr_{X}1 = 1

(iii) Tr_{X}^{2} = Tr_{X}

(iv) Tr_{X}(afb) = aTr_{X}(f)b , for any a,b ∈ A_{ℤ^{d}\X}

We note that in case of classical potential Φ the restriction of the state ω_{Φ} to the elements of the classical subalgebra f ∈ C can be identified with the Gibbs measure as follows

ω_{Φ}(f) = μ_{Φ}(f_{c})

With the state ω_{Φ} we associate the following (interpolating family) of Banach spaces 𝕃_{p}(ω_{Φ}, \frac{1}{2}), p ∈ (1, ∞] defined as a closure of A with respect to the corresponding norms given by

||F ||_{p}^{p} ≡ F- lim Tr|(ρ^{(Λ)})^{\frac{1}{2p}} f(ρ^{(Λ)})^{\frac{1}{2p}} |^{p}

Note that each 𝕃_{p}(ω_{Φ}) when restricted to the classical subalgebra C coincides with the classical L_{p}(μ_{Φ}) space associated to the Gibbs measure μ_{Φ}.

For later purposes we introduce an extension of the classical conditional expectations described in the following statement.

Proposition: For every classical potential Φ ∈ B_{1} and any X ⊂⊂ ℤ^{d} the following generalized conditional expectation is well defined

E_{X,Φ}(f) = Tr_{X}(γ_{X}^{∗}fγ_{X})  (3)

with

γ_{X} = γ_{X}^{∗} ≡ e^{− \frac{1}{2} U_{X}}(Tr_{X}e^{−U_{X}})^{− \frac{1}{2}} ∈ C ⊂ A  (4)

where U_{X} ≡ ∑_{Y ∩X≠∅} Φ_{Y}. If the potential Φ is of finite range we have γ_{X} ∈ A_{loc}.

Remark : For a description of a more general class of the generalised conditional expectations see Appendix I.

Exercise : Prove the above statement. Show that the generalised conditional expectations preserve unit and positivity (as in case of partial trace properties (i) and (ii) above), but in general do not have the projection property (contrary to the partial trace property (iii); the property (iv) is not true either).

Note that on the classical subalgebra C the generalized conditional expectation E_{X} introduced in the proposition above coincides with the conditional expectation of a Gibbs state for the same potential.

Using the generalised conditional expectations we define a finite volume jump type Markov generator

L^{X,Λ} = ∑_{j∈Λ} L_{X+j}  (5)

with

L_{X+j}f ≡ E_{X+j,Φ}(f) − f  (6)

where X + j denotes the translation of a finite set X by a vector j ∈ ℤ^{d}. In a special case when Φ ≡ 0 , we get a free Poissonian generator Tr_{X+j}f − f ≡ −∂_{X+j}f . (The particular case of this notation includes the operator ∂_{j} which is called a discrete gradient and will be frequently used later on. )

Exercise : Prove that L_{X+j} are symmetric in 𝕃_{2}(ω_{Φ}).

Given a potential Ψ ∈ B_{1} and a set Λ ⊆ ℤ^{d} , one can also introduce a possibly unbounded densely defined derivation

δ_{Ψ}^{Λ}(f) ≡ ∑_{X⊂⊂Λ} δ_{Ψ_{X}}(f) ≡ ∑_{X⊂⊂Λ} i[Ψ_{X}, f]

(for Λ = ℤ^{d} the corresponding superscript will be omitted).

Exercise : Show that for a nonabelian algebra, even if Ψ = Φ the corresponding operators L^{X,Λ} and δ_{Ψ}^{Λ} do not commute. (Hint : Consider action of the commutator on an element of A_{j} ≡ A_{{j}} for some j ∈ ℤ^{d}.)

To construct a dissipative dynamics, later on we will need the potentials of class 𝔹_{2} , that is those for which the following norm is finite.

||Φ||_{2} ≡ sup_{j∈ℤ^{d}} ∑_{Y ∋j} ∑_{k∈Y} ||∂_{k}Φ_{Y} || < ∞

We will also use the following (possibly bigger) norm

||Φ||˜2 ≡ sup_{k∈ℤ^{d}} ∑_{j∈ℤ^{d}} ∑_{l∈ℤ^{d}: d(l,j)≧d(k,j)} ∑_{Y ∋j,l} ||∂_{l}Φ_{Y} || < ∞  (7)

and denote the corresponding space by 𝔹˜2 . We will show the following result

Theorem: Let Ψ ∈ 𝔹_{2} and let Φ ∈ 𝔹˜2 be a classical potential. Let L_{Φ}^{X,Λ} be a finite volume jump type generator corresponding to the generalized conditional expectations E_{X+j,Φ} , for a given finite set X ⊂⊂ ℤ^{d} and j ∈ Λ ⊂⊂ ℤ^{d}, and let P_{t}^{X,Φ,Ψ,Λ} ≡ e^{t(L_{Φ}^{X,Λ}+δ_{Ψ}^{Λ})}. Then the infinite volume semigroup

P_{t}^{X,Φ,Ψ} ≡ F- lim P_{t}^{X,Φ,Ψ,Λ}  (8)

is a well defined unit and positivity preserving semigroup with a Feller property, that is

P_{t}^{X,Φ,Ψ}(A) ⊆ A.  (9)

We stress that the potential Ψ in the above is not assumed to be classical and thus one obtains a truly new dissipative semigroups which is not possible to have in the commutative context. In general in this case one does not know a’priori the invariant states. If Φ = Ψ , the hamiltonian term cannot be detected on the classical subalgebra. Moreover the set of invariant states contains all the Gibbs states, (in general we can have many of them). The corresponding semigroup when restricted to the classical subalgebra coincides with the Glauber dynamics for the classical spins. We note that even in this case the semigroup with hamiltonian term is nontrivial; ( that is the generator L^{X} does not commute with the hamiltonian automorphism). We remark that by weak compactness of A , the property (9) implies the existence of P_{t}^{X,Φ,Ψ}-invariant state. Postponing the construction of the dissipative dynamics to the next section, we discuss first the corresponding ergodicity properties.

Strong Ergodicity

For k ∈ ℤ^{d} , we define a discrete gradient ∂_{k}f ≡ f−Tr_{k}f and using it introduce the following seminorm (frequently called a triple bar norm)

|||f||| ≡ ∑_{k∈ℤ^{d}} ||∂_{k}f||

One can see that this seminorm vanishes only on the multiples of the unit operator. We will use A_{1} to denote the set of all elements in A for which the triple bar norm is finite. With this notation we have the following ergodicity result for the semigroup P_{t}^{X,Φ,Ψ} introduced above.

Theorem : Given a finite set X ⊂⊂ ℤ^{d}, there is β_{0} > 0 and λ_{0} ≡ λ_{0}(β_{0}) > 0 such that for any ||Φ||˜2 < β_{0} and ||Ψ||_{2} < λ_{0} , we have

|||P_{t}^{X,Φ,Ψ}(f)||| ≤ e^{−mt}|||f|||  (10)

with a constant m ∈ (0, ∞) independent of f .

This result implies the following immediate corollary.

Corollary: Given a finite set X ⊂⊂ ℤ^{d} , there is β_{0} > 0 and λ_{0} ≡ λ_{0}(β_{0}) > 0 such that for any ||Φ||˜2 < β_{0} and ||Ψ||_{2} < λ_{0} there exists a unique P_{t}^{X,Φ,Ψ} - invariant state ω for which we have

||P_{t}^{X,Φ,Ψ}f−ωf|| ≦ 2e^{−mt}|||f|||

with a constant m ∈ (0, ∞) independent of f ∈ A_{1}.

Proof of the Exponential Decay to Equilibrium in the Triple Bar Norm

Let P_{t}^{X} ≡ P_{t}^{X,Φ,Ψ} denote the semigroup corresponding to the generator L^{X} + δ_{Ψ} with L^{X} ≡ L_{Φ}^{X} ≡ ∑_{j∈ℤ^{d}} L_{X+j} . To show the ergodicity in the triple bar norm we need to study the behaviour of ||∂_{k}P_{t}^{X}f|| for all j ∈ ℤ^{d} . To this end we set L^{X,k} ≡ L^{X} − ∑_{j:X+j∋k} L_{X+j}, and define an auxiliary semigroup P_{t}^{X,k} corresponding to the generator L^{X,k} + δ_{Ψ} . We will use this auxiliary semigroup to get the desired bound as follows. We note that for s ∈ [0, t) , we have

\frac{d}{ds}e^{s|X|}P_{t−s}^{X,k}∂_{k}P_{s}^{X}f=e^{s|X|}P_{t−s}^{X,k} ([∂_{k}, L^{X,k}]+[∂_{k}, δ_{Ψ}]) P_{s}^{X}f  (11)

where one uses the definition of the corresponding generators together with the fact that for k∈X + j we have ∂_{k}L_{X+j}F = −∂_{k}F . Hence, integrating from zero to t and dividing both sides by e^{s|X|} , we obtain

∂_{k}P_{t}^{X}f = e^{−t|X|}P_{t}^{X,k}∂_{k}f+ ∫_{0}^{t} ds \frac{d}{ds}e^{(s−t)|X|}P_{t−s}^{X,k}∂_{k}P_{s}^{X}f
= e^{−t|X|}P_{t}^{X,k}∂_{k}f + ∫_{0}^{t} ds e^{(s−t)|X|}P_{t−s}^{X,k} ([∂_{k}, L^{X,k}] + [∂_{k}, δ_{Ψ}]) P_{s}^{X}f

Because P_{t}^{X,k} is a Markov semigroup, we get the following norm estimate.

||∂_{k}P_{t}^{X}f|| ≤ e^{−t|X|} ||∂_{k}f|| +  (12)

+ ∫_{0}^{t} ds e^{(s−t)|X|} ∑_{j:k∉X+j} ||[∂_{k}, L_{X+j}]P_{s}^{X}f|| +

+ ∫_{0}^{t} ds e^{(s−t)|X|} ∑_{Y: Y ∋k} ||[∂_{k}, δ_{Ψ_{Y}}]P_{s}^{X}f||

We show below (see Proposition R4 below) that for k ∉ X + j

∑_{j:k∉X+j} ||[∂_{k}, L_{X+j}]f|| ≤ ∑_{j: k∉X+j} ∑_{l∈ℤ^{d}}a_{kl}^{j}||∂_{l}f||  (13)

with some constants a_{ki}^{j} ≥ 0 such that

sup_{l∈ℤ^{d}} ∑_{k} ∑_{j: k∉X+j} a_{kl}^{j} ≦ ε|X|  (14)

for some ε ≡ ε_{Φ} ∈ (0, 1) provided ||Φ||_{2} is sufficiently small. Then the sum over k of the first integral term on the right hand side of (12), can be bounded by

∑_{k∈ℤ^{d}} ∑_{j: k∉X+j} ||[∂_{k}, L_{X+j}]P_{s}^{X}f|| ≦ ε|X| · |||P_{s}^{X}f|||  (15)

On the other hand, we claim that , (see Lemma R1 proven later),

∑_{k∈ℤ^{d}} ∑_{Y: Y ∋k} ||[∂_{k}, δ_{Ψ_{Y}}]P_{s}^{X}f|| ≦ λ|||P_{s}^{X}f|||  (16)

with some nonnegative constant λ ≦ 2||Ψ||_{2} . Summing inequalities (12) over all k ∈ ℤ^{d} and using (15)-(16), we arrive at the following inequality.

|||P_{t}^{X}f||| ≤ e^{−t|X|}|||f||| + (ε|X| + λ) ∫_{0}^{t} ds e^{(s−t)|X|}|||P_{s}^{X}f||| (17)

The inequality (17) can be solved by iteration and, if ε ∈ (0, 1) and λ ∈ (0, λ_{0}) with λ_{0} ≡ (1 − ε)|X|/2||Ψ||_{2} , one gets the following exponential decay to equilibrium

|||P_{t}^{X}f||| ≤ e^{−mt}|||f|||

m = (1 − ε)|X| − 2||Ψ||_{2}  (18)

Proof of Corollary : We note first that, by weak compactness of the space of states on A and the fact that by our construction P_{t}^{X} ≡ P_{t}^{X,Φ,Ψ} has Feller property, the set of P_{t}^{X}-invariant states is nonempty. Let ω be one such state. Then we have

||P_{t}^{X}f − ωf|| = ||P_{t}^{X}f − ω(P_{t}^{X}f)|| = ||Θ (P_{t}^{X}f ⊗ 1 − 1 ⊗ P_{t}^{X}f) ||
≤ ||P_{t}^{X}f ⊗ 1 − 1 ⊗ P_{t}^{X}f||

the norm on the right hand side means the norm on the injective tensor product algebra A by itself, while Θ is a completely positive map from A⊗A to A defined by Θ(x_{1} ⊗ x_{2}) ≡ ω(x_{1})x_{2}, where x_{1}, x_{2} ∈ A . Choosing some lexicographic sequence {j_{n}}_{n∈N} in ℤ^{d} and observing that

P_{t}^{X}f ⊗ 1 − 1 ⊗ P_{t}^{X}f = Σ ⊗ 1 − 1 ⊗ Σ

with

Σ ≡ (P_{t}^{X}f − Tr_{j_{1}}P_{t}^{X}f) + ∑_{n∈ℕ}Tr_{{j_{1},..,j_{n}}}(P_{t}^{X}f − Tr_{j_{n+1}}P_{t}^{X}f)
= ∂_{j_{1}}P_{t}^{X}f + ∑_{n∈ℕ}Tr_{{j_{1},..,j_{n}}}∂_{j_{n+1}}P_{t}^{X}f

one easily arrives at the following inequality

||P_{t}^{X}f − ωf|| ≤ 2|||P_{t}^{X}f|||

This together with the above discussed ergodicity with respect to the triple bar norm concludes the proof.

Construction of Jump Type Dynamics

In this section we construct a Markov semigroup on A as a limit of a sequence of semigroups P_{t}^{i} ≡ e^{tŁ_{i}} ≡ P_{t}^{X,Λ_{i}} , defined for Λ_{i} ∈ F by bounded generators.

To control the convergence of the sequence of interest to us we study the difference of consecutive elements acting on local elements as follows. We have

P_{t}^{2}f − P_{t}^{1}f = ∫_{0}^{t} ds\frac{d}{ds}P_{t−s}^{1}P_{s}^{2}f = ∫_{0}^{t} dsP_{t−s}^{1}(Ł_{2} − Ł_{1})P_{s}^{2}f  (19)

Hence, using the contractivity property of Markov semigroup, we get the following norm bound

||P_{t}^{2}f − P_{t}^{1}f|| = || ∫_{0}^{t} dsP_{t−s}^{1}(Ł_{2} − Ł_{1})P_{s}^{2}f|| ≤ ∫_{0}^{t} ds||(Ł_{2} − Ł_{1})P_{s}^{2}f||(20)

To bound the right hand side we need to study the expression (Ł_{2} − Ł_{1})P_{s}^{2}f. As the difference of two Markov operators of interest to us equals to a sum of elementary generators L_{X+j} and δ_{Ψ_{Y}}, it is sufficient to study L_{X+j}P_{s}^{2}f for j ∈ ℤ^{d} and δ_{Ψ_{Y}} (P_{s}^{2}f). We will show later that the following regularity property is true, (see Proposition R3 below),

||L_{X+j}F || ≤ ∑_{k} b_{jk}||∂_{k}F ||  (R)

with some constants b_{jk} ≡ b_{jk}^{X} ∈ [0, ∞) such that sup_{j} ∑_{k} b_{jk} < ∞ . Naturally, since δ_{Ψ_{Y}} (F) is proportional to [Ψ_{Y} , F], similar regularity bound for the hamiltonian part is true (see Lemma R1 below). Therefore we need to study the behavior of ∂_{k}P_{t}^{2}f. To this end we use the following arguments. We note that

∂_{k}P_{s}^{2}f = P_{s}^{2}∂_{k}f + ∫_{0}^{s} ds̃ \frac{d}{ds̃}P_{s−s̃}^{2}∂_{k}P 2 ; s̃f
= P_{s}^{2}∂_{k}f + ∫_{0}^{s} ds̃ P_{s−s̃}^{2}[∂_{k}, Ł_{2}]P 2 ; s̃f

and hence using contractivity property of the Markov semigroups, we get

||∂_{k}P_{s}^{2}f|| ≤ ||∂_{k}f|| + ∑_{i∈Λ_{2}} ∫_{0}^{s} ds̃||[∂_{k}, L_{X+i}]P 2 ; s̃f||  (21)

+ ∑_{Y ⊂Λ_{2}} ∫_{0}^{s} ds̃||[∂_{k}, δ_{Ψ_{Y}}]P 2 ; s̃f||

Later we will show that the following condition holds, (see Proposition R4),

∑_{i∈ℤ^{d}} ||[∂_{k}, L_{X+i}]F || ≦ ∑_{l} G_{X+i}(k, l)||∂_{l}F ||  (C)

with some matrix G_{X+i}(k, l) such that

sup_{l∈ℤ^{d}} ∑_{k∈ℤ^{d}} ∑_{i∈ℤ^{d}} G_{X+i}(k, l) ≤ κ|X| < ∞

We set

G(k − l) ≡ G_{X}(k − l) ≡ ∑_{i∈ℤ^{d}} G_{X+i}(k, l)  (22)

A similar bound follow from Lemma R1 for the hamiltonian part. Denoting by G the sum of matrix (22) and the corresponding one for the hamiltonian part, we can bound the right hand side of (21) by

||∂_{k}P_{s}^{2}f|| ≤ ||∂_{k}f|| + ∫_{0}^{s} ds̃∑_{l} G(k − l)||∂_{l}P 2 ; s̃f||  (23)

Solving the inequality (23) by iteration we obtain the following bound

||∂_{k}P_{s}^{2}f|| ≤ ∑_{l} (e^{sG})_{kl} ||∂_{l}f||  (24)

Combining (19), (24), we arrive at the estimate which has the following form

||P_{t}^{2}f − P_{t}^{1}f|| ≤ t ∑_{j∈Λ_{2}\Λ_{1}} ∑_{k,l} b_{jk} (e^{tG})_{kl} ||∂_{l}f||  (25)

with the appriopriate coefficients b_{jk} . Hence for any Λ_{2} ⊂ ℤ^{d} containing a set Λ_{1} we have

||P_{t}^{2}f − P_{t}^{1}f|| ≤ t ∑_{j∈ℤ^{d}\Λ_{1}} ∑_{k,l} b_{jk} (e^{tG})_{kl} ||∂_{l}f||  (26)

Using the summability properties of b_{jk} and G, one concludes that the limit

P_{t}^{X}f ≡ F- lim P_{t,Λ}^{X}f

exists for all local elements f ∈ A_{loc}. Hence, by continuity in the norm || · ||, it exists also for any f ∈ A.

Exercise : Exponential Approximation Property

• Using (26) show the following property (which will be useful later in Lecture II). For f ∈ A_{loc} we use Λ(f) the smallest set Λ ⊂ ℤ^{d} such that one has f ∈ A_{Λ} .

Theorem : Let P_{t}^{X,Λ} be a finite dimensional jump type semigroup for a finite range classical potential. Then for any A ∈ ℝ^{+}, there exists a constant B depending only on A such that for any sufficiently large finite set Λ ⊂ ℤ^{d} containing Λ(f) and satisfying dist(Λ(f), Λ^{c}) ≥ Bt, one has

||P_{t}^{X}f − P_{t}^{X,Λ}f||_{∞} ≤ e^{−At}|||f|||.  (27)

• Using the estimate (26) generalise the above approximation property replacing the linear function Bt with the appropriate functions φ(t) . Show that by choosing φ(t) ≡ e^{εt} with some positive ε one can include a comprehensive class of classical long range potentials.

Regularity Properties of the Elementary Markov Generators

In this Section we study some regularity properties of derivations δ_{Ψ} and the elementary operators L_{X+j} . (Below we use a notation χ(A) for a charactersitc function of a set A . )

Lemma R1 : For any Y ⊂⊂ ℤ^{d} and any j ∈ Y

||[∂_{j}, δ_{Ψ_{Y}}]f|| ≦ ∑_{k∈Y} (2χ(j, k ∈ Y)||∂_{j}Ψ_{Y} ||) · ||∂_{k}f||  (28)

and hence

∑_{j∈ℤ^{d}} ||[∂_{j}, δ_{Ψ}]f|| ≦ ∑_{j∈ℤ^{d}} ∑_{Y: Y ∋j} ||[∂_{j}, δ_{Ψ_{Y}}]f||

≤ 2 sup_{k∈ℤ^{d}} ( ∑_{Y: Y ∋k} ∑_{j: Y ∋j} ||∂_{j}Ψ_{Y} ||) · |||f|||

Proof of Lemma R1 : Using the definition of the derivation and the discrete gradient ∂_{j} ≡ I − Tr_{j} , we have

−i[∂_{j}, δ_{Ψ_{Y}}]f = ∂_{j}([Ψ_{Y} , f]) − [Ψ_{Y} , ∂_{j}f]  (29)
= [∂_{j}Ψ_{Y} , Tr_{j}f] − Tr_{j}[∂_{j}Ψ_{Y} , ∂_{j}f]

Next we note that , because Ψ_{Y} ∈ A_{Y} and Tr_{Y} f∈A_{ℤ^{d}\Y} , we have

[∂_{j}Ψ_{Y} , Tr_{j}f] =[∂_{j}Ψ_{Y} , Tr_{j}f−Tr_{Y}Tr_{j}f] = ∑_{l=1,..,|Y |}[∂_{j}Ψ_{Y} , Tr_{Y_{l−1}}∂_{k_{l}}f] (30)

where Y_{0} ≡ {j} and Y_{l} ≡ Y_{l−1} ∪ {k_{l}} , l = 1, .., |Y | , with some lexicographic ordering of the set Y \{j} . Hence

||[∂_{j}, δ_{Ψ_{Y}}]f|| ≦ 2χ(j ∈ Y)||∂_{j}Ψ_{Y} || ∑_{k∈Y} ||∂_{k}f||  (31)

with χ(j ∈ Y) equal to one if j ∈ Y and zero otherwise. Since

[∂_{j}, δ_{Ψ}]f= ∑_{Y: Y ∋j}[∂_{j}, δ_{Ψ_{Y}}]f  (32)

using (31) we arrive at

∑_{j∈ℤ^{d}} ||[∂_{j}, δ_{Ψ}]f|| ≦ ∑_{j∈ℤ^{d}} ∑_{Y: Y ∋j} ||[∂_{j}, δ_{Ψ_{Y}}]f|| ≦  (33)

≦ 2 (∑_{j∈ℤ^{d}} ∑_{Y: Y ∋j} ||∂_{j}Ψ_{Y} || ∑_{k∈Y} ||∂_{k}f||) = 2 ∑_{k∈ℤ^{d}} ||∂_{k}f|| ( ∑_{Y: Y ∋k} ∑_{j: Y ∋j} ||∂_{j}Ψ_{Y} ||)

≦ 2 sup_{k∈ℤ^{d}} ( ∑_{Y: Y ∋k} ∑_{j: Y ∋j} ||∂_{j}Ψ_{Y} ||) · |||f|||

which ends the proof of the lemma.

Recall that the elementary Markov generator is defined by

L_{X+j}(f) = Tr_{X+j}(γ_{X+j}^{∗}fγ_{X+j}) − f  (34)

We assume that γ_{X+j} ∈ A and satisfies Tr_{X+j}(γ_{X+j}^{∗}γ_{X+j}) = 1. These assumptions assure that the finite volume dynamics P_{t}^{X,Λ} ≡ e^{tL_{Λ}^{X}} is well defined and has the Feller property. For later purposes we note that the following representation is true.

Lemma R2 : For any finite set X, the operator L_{X} admits the following representation

L_{X}(f) = \frac{1}{2}Tr_{X} ([γ_{X}^{∗}, f]γ_{X}) + \frac{1}{2}Tr_{X} (γ_{X}^{∗}[f, γ_{X}])

+ \frac{1}{2}Tr_{X} (\{(f − Tr_{X}f), (γ_{X}^{∗}γ_{X} − 1)\})

+(Tr_{X}f − f)  (35)

and so

||L_{X}(f)|| ≤ \frac{1}{2} (||[γ_{X}^{∗}, f]|| · ||γ_{X}|| + ||γ_{X}^{∗}|| · ||[f, γ_{X}]||)  (36)

+ (||γ_{X}^{∗}γ_{X} − 1|| + 1) ∑_{k∈X} ||∂_{k}f||

Exercise : Prove Lemma R2 . Note that

L_{X}(f) = Tr_{X}(γ_{X}^{∗}fγ_{X}) − f  (37)
= \frac{1}{2}Tr_{X} ([γ_{X}^{∗}, f]γ_{X}) + \frac{1}{2}Tr_{X} (γ_{X}^{∗}[f, γ_{X}]) + \frac{1}{2}Tr_{X} (\{f, γ_{X}^{∗}γ_{X}\}) − f

Next use the normalisation condition T r_{X}(γ_{X}^{∗}γ_{X}) = 1 and a property of the partial trace to represent the last part of the right hand side of (37) as follows

\frac{1}{2}Tr_{X} (\{f, γ_{X}^{∗}γ_{X}\}) − f = \frac{1}{2}Tr_{X} (\{(f − Tr_{X}f), (γ_{X}^{∗}γ_{X} − 1\})) + (Tr_{X}f − f)

(38)

Combining this with (37) we get the desired representation (35) . The inequality (36) easily follows from (35) .

Using the above lemma we obtain the following condition for the regularity.

Proposition R3: Suppose

sup_{j∈ℤ^{d}} ∑_{k∈ℤ^{d}} ||∂_{k}γ_{X+j}|| < ∞  (39)

Then the operators L_{X+j} given by (34) satisfies the following regularity condition

||L_{X+j}(f)|| ≤ ∑_{k∈ℤ^{d}} b_{jk} ||∂_{k}f||  (40)

with some nonnegative constants b_{jk} ≡ b_{jk}^{X} such that sup_{j,k∈ℤ^{d}} b_{jk} < ∞ . Additionally if

sup_{k∈ℤ^{d}} ∑_{j∈ℤ^{d}} ∑_{l: d(l,j)≧d(k,j)}||∂_{l}γ_{X+j}|| < ∞  (41)

then the operator

L ≡ ∑_{j∈ℤ^{d}} L_{X+j}

is well defined on a dense domain containing A_{1} .

Exercise : Show that in case of commutative algebra, (using T r_{X}(γ_{X}^{∗}γ_{X}) = 1) one gets

L_{X}(f) = Tr_{X}(γ_{X}^{∗}fγ_{X}) − f = Tr_{X}(γ_{X}^{∗}(f − Tr_{X}f)γ_{X}) − (f − Tr_{X}f)

and therefore

||L_{X}(f)|| ≦ 2 ∑_{k∈X} ||∂_{k}f||

whence

|| ∑_{j∈ℤ^{d}} L_{X+j}(f)|| ≦ 2|X| · |||f|||

Thus no additional conditions on interaction in commutative case is necessary.

Proof of Proposition R3 : In view of Lemma R2, inequality (36), to prove (40) it is sufficient to prove that the following inequality is true

||[γ_{X+j}, f]|| ≤ ∑_{k} c_{jk} ||∂_{k}f||  (42)

with some nonnegative constants c_{jk} ≡ c_{jk}^{X} such that sup_{k} c_{jk} < ∞ . To this end we introduce a lexicographic order l_{m} ∈ ℤ^{d}, m ∈ ℕ, with l_{1} ≡ j and define a sequence of sets Y_{m+1} = Y_{m} ∪ {l_{m+1}}, Y_{0} ≡ ∅ . One may and do assume that the lexicographic order has the following property

m ≧ n ⟹ d(l_{m}, j) ≧ d(l_{n}, j)

for the d(x, y) ≡ max_{q=1,..,d} |x^{(q)} − y^{(q)}| . Since a commutator vanishes if one of its entries equals to a multiple of the identity, we have the following representation .

[g, f] = ∑_{m∈ℕ}[Tr_{Y_{m+1}^{c}}(∂_{l_{m}}g), f] = ∑_{m∈ℕ} ∑_{n∈ℕ}[Tr_{Y_{m}^{c}}(∂_{l_{m}}g), Tr_{Y_{n−1}}∂_{l_{n}}f] (43)

with a convention in the second term that Tr_{∅} ≡ I. (Note that the last partial trace on the right hand side of (43) is associated to the set Y_{n−1}, not to its complement, as in the previous equality.) Using also the local structure of our algebra, we get

[g, f] = ∑_{n∈ℕ} ∑_{m∈ℕ: m+1≧n}[Tr_{Y_{m}^{c}}(∂_{l_{m}}g), Tr_{Y_{n−1}}(∂_{l_{n}}f)]  (44)

Hence we obtain

||[g, f]|| ≤ 2 ∑_{n∈ℕ} ( ∑_{m∈ℕ: m+1≧n}||∂_{l_{m}}g||) · ||∂_{l_{n}}f||  (45)

An application of this formula with g = γ_{X+j} leads us to the inequality (42) with the corresponding constants c_{jk} given by

c_{jk} = 2 ∑_{l_{m}: d(l_{m}, j)≥d(k, j)} ||∂_{l_{m}}γ_{X+j}||  (46)

Thus the condition (39) implies that

sup_{j,k∈ℤ^{d}} c_{jk} < ∞  (47)

Similar considerations involving γ_{X+j}^{∗} together with the use of (35)-(36) and (42)-(47) allows us to construct the constants b_{jk} ≡ b_{jk}^{X} such that

||L_{X+j}(f)|| ≤ ∑_{k∈ℤ^{d}} b_{jk} ||∂_{k}f||  (48)

and

sup_{j,k∈ℤ^{d}} b_{jk} < ∞  (49)

If (41) is satisfied, then

sup_{k∈ℤ^{d}} ∑_{j∈ℤ^{d}} b_{jk} < ∞  (50)

and one can sum over j ∈ ℤ^{d} the inequalities (48). This implies that L ≡ ∑_{j} L_{X+j} is well defined on the set A_{1} of all elements of the algebra which have finite triple bar norm. This ends the proof of Proposition R3.

Exercise : Show that if γ_{X+j} ≡ γ_{X+j,Φ} for a classical potential Φ, then the condition

||Φ||_{2} ≡ sup_{j∈ℤ^{d}} ∑_{Y ∋j} ∑_{k∈Y} ||∂_{k}Φ_{Y} || < ∞  (51)

implies

sup_{j∈ℤ^{d}} ∑_{k∈ℤ^{d}} ||∂_{k}γ_{X+j}|| < ∞

Moreover if

||Φ||˜2 ≡ sup_{k∈ℤ^{d}} ∑_{j∈ℤ^{d}} ∑_{l∈ℤ^{d}: d(l,j)≧d(k,j)} ∑_{Y ∋j,l} ||∂_{l}Φ_{Y} || < ∞  (52)

then one also has

sup_{k∈ℤ^{d}} ∑_{j∈ℤ^{d}} ∑_{l: d(l,j)≧d(k,j)}||∂_{l}γ_{X+j}|| < ∞

Next we shall study the 2nd regularity condition. Proposition R4 : Suppose

sup_{k∈ℤ^{d}} ∑_{j∈ℤ^{d}} ∑_{l: d(l,j)≧d(k,j)}||∂_{l}γ_{X+j}|| < ∞  (53)

Then for k ∉ X + j , we have

||[∂_{k}, L_{X+j}](f)|| ≤ ∑_{l∈X+j} a_{kl}^{X+j} · ||∂_{l}f||  (54)

with

a_{kl}^{X+j} ≤ ||∂_{k}γ_{X+j}|| · (b_{0,l} + χ(l ∈ X + j)||γ_{X+j}||)

+ ∑_{{lm: d(lm,X+j) ≥ d(l,X+j) \atop d(lm,X+j) ≥ d(k,X+j)}} ||γ_{X+j}|| · ||∂_{l_{m}}γ_{X+j}^{∗}||

Proof: Using Lemma R2 one can see that for any k ∈ ℤ^{d} we have

[∂_{k}, L_{X}](f) = 𝔸_{1} + 𝔸_{2} + 𝔸_{3}  (55)

where

𝔸_{1}(f) ≡ \frac{1}{2} (Tr_{X} ([γ_{X}^{∗}, Tr_{k}f]γ_{X}) − Tr_{k}Tr_{X} ([γ_{X}^{∗}, f]γ_{X}))  (56)
𝔸_{2}(f) ≡ 𝔸_{1}(f^{∗})^{∗}  (57)

and

𝔸_{3}(f) ≡ \frac{1}{2}(Tr_{X} (\{(Tr_{k}f − Tr_{X}Tr_{k}f), (γ_{X}^{∗}γ_{X} − 1)\})
−Tr_{k}Tr_{X} ({(f − Tr_{X}f), (γ_{X}^{∗}γ_{X} − 1)}) )

Let us consider first 𝔸_{1}. After simple calculations one gets

2𝔸_{1} = Tr_{X} ([∂_{k}γ_{X}^{∗}, Tr_{k}f]γ_{X}) + Tr_{X} ([Tr_{k}γ_{X}^{∗}, Tr_{k}f]∂_{k}γ_{X})

− Tr_{k}Tr_{X} ([∂_{k}γ_{X}^{∗}, f]γ_{X}) − Tr_{k}Tr_{X} ([Tr_{k}γ_{X}^{∗}, f]∂_{k}γ_{X}) (58)

Now we use similar idea as in (43) - (45) to expand γ_{X}^{∗} and f. By this we get the following representation of the first term on the right hand side of (58)

Tr_{X} ([∂_{k}γ_{X}^{∗}, Tr_{k}f]γ_{X}) = ∑_{l_{m}} ∑_{l_{n}} Tr_{X} ([∂_{k}Tr_{Y_{m}^{c}}∂_{l_{m}}γ_{X}^{∗}, Tr_{k}Tr_{Y_{n−1}}∂_{l_{n}}f]γ_{X}) =

= ∑_{l_{n}} ∑_{{lm: d(lm,X) ≥ d(ln,X) \atop d(lm,X) ≥ d(k,X)}} Tr_{X} ([∂_{k}Tr_{Y_{m}^{c}}∂_{l_{m}}γ_{X}^{∗}, Tr_{k}Tr_{Y_{n−1}}∂_{l_{n}}f]γ_{X}) (59)

Hence we obtain the following bound on the first term on the right hand side of (58)

||Tr_{X} ([∂_{k}γ_{X}^{∗}, Tr_{k}f]γ_{X}) || ≤ 2 ∑_{l_{n}} ( ∑_{{lm: d(lm,X) ≥d(ln,X) \atop d(lm,X) ≥d(k,X)}} ||γ_{X}|| · ||∂_{l_{m}}γ_{X}^{∗}||) · ||∂_{l_{n}}f|| (60)

The similar estimate will remain true also for the third term on the right hand side of (58). The second (as well as the last) term from the right hand side of (58) can be bounded as follows

||Tr_{X} ([Tr_{k}γ_{X}^{∗}, Tr_{k}f]∂_{k}γ_{X}) || ≤ 2 ∑_{l∈ℤ^{d}}||∂_{k}γ_{X}|| · b_{0,l}^{X} ||∂_{l}f||  (61)

Combining (60) and (61) we get

||𝔸_{1} + 𝔸_{2}|| ≤ 2 ∑_{l∈ℤ^{d}} (||∂_{k}γ_{X}|| · b_{0,l}^{X} + ∑_{{lm: d(lm,X) ≥ d(l,X) \atop d(lm,X) ≥ d(k,X)}} ||γ_{X}|| · ||∂_{l_{m}}γ_{X}^{∗}||) · ||∂_{l}f|| (62)

For the term 𝔸_{3} one has

2𝔸_{3} ≡ Tr_{X}({(Tr_{k}f − Tr_{X}Tr_{k}f), ∂_{k}(γ_{X}^{∗}γ_{X})})
−Tr_{k}Tr_{X}({(f − Tr_{X}f), ∂_{k}(γ_{X}^{∗}γ_{X})})

whence we obtain

||𝔸3|| ≤ ∑_{l∈X}(3||γ_{X}|| · ||∂_{k}γ_{X}||) · ||∂_{l}f||  (63)

Combining (62) and (63), we obtain

||[∂_{k}, L_{X}](f)|| ≤ ∑_{l∈X} a_{kl} · ||∂_{l}f||  (64)

with

a_{kl} ≤ 2 (||∂_{k}γ_{X}|| · b_{0,l}^{X} + ∑_{{lm: d(lm,X) ≥ d(l,X) \atop d(lm,X) ≥ d(k,X)}} ||γ_{X}|| · ||∂_{l_{m}}γ_{X}^{∗}||)
+ χ_{(l∈X)} 3||γ_{X}|| · ||∂_{k}γ_{X}||

This ends the proof of Proposition R4.

Equivalence of Quadratic Forms and 𝕃_{2} Ergodicity

Above we have constructed an explicit class of dissipative dynamics for quantum spin systems on a lattice with strong ergodic properties. In general if we choose Φ ≠ Ψ, we may not know a’priori the (unique) invariant state.

In case of commutative spin systems of classical statistical mechanics in general one can get ergodicity for the larger range of temperatures when choosing the elementary generators L_{X} to be defined with a cube X of large size, see [3]. In the that situation there is a standard way to use this strong ergodicity result for dynamics of block spin flips type with some large block X (without hamiltonian term), to prove the 𝕃_{2} ergodicity of all the other block spin flip generators. The main idea of this is based on the equivalence of Dirichlet forms associated to generators L^{Y} with Y ⊂⊂ ℤ^{d}. As shown in Appendix I, this equivalence remains true in the considered case of quantum block spin flip generators associated to finite range classical potentials.

Theorem : For every X, Y ⊂⊂ ℤ^{d} there is a constant C_{X,Y} ∈ (0, ∞) such that

C_{X,Y}^{−1} ⟨f, −L_{Φ}^{X}f⟩_{𝕃_{2}(ω_{Φ}, \frac{1}{2} )} ≤ ⟨f, −L_{Φ}^{Y}f⟩_{𝕃_{2}(ω_{Φ}, \frac{1}{2} )} ≤ C_{X,Y} ⟨f, −L_{Φ}^{X}f⟩_{𝕃_{2}(ω_{Φ}, \frac{1}{2} )}(65)

If for some sufficiently large block X the corresponding dynamics decays exponentially to equilibrium in the algebra norm, then - (since by our construction the corresponding generator is selfadjoint in 𝕃_{2}(ω_{Φ}, \frac{1}{2})) - one can apply spectral theory together with the strong ergodicity to conclude the existence of the spectral gap at the top of the spectrum of the generator . From this and the equivalence of the Dirichlet forms one concludes the existence of the spectral gap for all the other generators of similar type. Thus it pays of having the dissipative dynamics defined by a selfadjoint generator as we get the following result.

Corollary : If for some X ⊂⊂ ℤ^{d} the corresponding dynamics P_{t}^{X} ≡ e^{tL_{Φ}^{X}} is strongly exponentially ergodic in the sense that

||P_{t}^{X}f − ω_{Φ}(f)|| ≦ e^{−mt}2|||f|||  (66)

with some m ∈ (0, ∞) , then for any Y ⊂⊂ ℤ^{d} the corresponding block spin flip generator L_{Φ}^{Y} as a selfadjoint operator in 𝕃_{2}(ω_{Φ}, \frac{1}{2}) has a spectral gap and the corresponding stochastic dynamics P_{t}^{Y} satisfies

||P_{t}^{Y} f − ω_{Φ}(f)|| _{𝕃_{2}(ω_{Φ}, \frac{1}{2} )} ≦ e^{−m_{Y} t}||f − ω_{Φ}(f)|| _{𝕃_{2}(ω_{Φ}, \frac{1}{2} )}  (67)

with some m_{Y} ∈ (0, ∞).

Proof: To show (67) note that

\frac{d}{dt}||P_{t}^{Y} f − ω_{Φ}(f)||_{𝕃_{2}(ω_{Φ}, \frac{1}{2} )}^{2} = ⟨P_{t}^{Y} f, L_{Φ}^{Y}P_{t}^{Y} f⟩_{𝕃_{2}(ω_{Φ}, \frac{1}{2} )}  (68)

By equivalence of Dirichlet forms we can bound the right hand side of (68) by

⟨P_{t}^{Y} f, L_{Φ}^{Y}P_{t}^{Y} f⟩_{𝕃_{2}(ω_{Φ}, \frac{1}{2} )} ≦ C_{X,Y}^{−1} ⟨P_{t}^{Y} f, L_{Φ}^{X}P_{t}^{Y} f⟩_{𝕃_{2}(ω_{Φ}, \frac{1}{2} )}  (69)

Next, since by (66) and spectral theory L_{Φ}^{X} satisfies the spectral gap inequality, we in particular have

⟨P_{t}^{Y} f, L_{Φ}^{X}P_{t}^{Y} f⟩_{𝕃_{2}(ω_{Φ}, \frac{1}{2} )} ≦ −m_{X}||P_{t}^{Y} f − ω_{Φ}(f)||_{𝕃_{2}(ω_{Φ}, \frac{1}{2} )}^{2}  (70)

with some m_{X} ≧ m > 0 . Combining this with (68) and (69), we obtain

\frac{d}{dt}||P_{t}^{Y} f − ω_{Φ}(f)||_{𝕃_{2}(ω_{Φ}, \frac{1}{2} )}^{2} ≦ −m_{X}C_{X,Y}^{−1} ||P_{t}^{Y} f − ω_{Φ}(f)||_{𝕃_{2}(ω_{Φ}, \frac{1}{2} )}^{2} (71)

Integration of this differential inequality yields (67) with

m_{Y} ≧ m_{X}C_{X,Y}^{−1}  (72)

Lecture II

Hypercontractivity and Strong Ergodicity

for Classical Systems

Introduction

In this lecture we provide a rapid introduction to the theory of hypercontractive semigroups in the commutative setting and describe another powerful strategy for studying ergodicity in infinite dimensional situation. This material will also serve as a primer and motivation for study of noncommutative extensions presented in the next two lectures. Let L_{p} ≡ L_{p}(μ) , p ∈ [1, ∞] , be the classical interpolating family associated to a probability measure μ (defined on some separable complete metric space).

A Markov semigroup P_{t} ≡ e^{tL} defined in L_{p} spaces, p ∈ [1, ∞] , is called (strictly) hypercontractive iff

||P_{t}f||_{q} ≦ ||f||_{2}  ((H))

with q ≡ 1 + e^{\frac{2t}{c}} , where c ∈ (0, ∞) , for all t ≧ 0 and all f ∈ L_{2} . Note that for t > 0 we have q > 2 . Then the Hölder inequality tells us that, without action of the semigroup, if at all f ∈ L_{q} , one has the opposite relation

||f||_{q} ≧ ||f||_{2}

In general, if t > 0 , for Markov semigroups one has only the following contractivity property for any p ∈ [1, ∞]

||P_{t}f||_{p} ≦ ||f||_{p}

Thus hypercontractivity is rather a very particular property. Similarly to the contractivity property, the hypercontractivity allows an infinitesimal description provided by the following Logarithmic Sobolev inequality

μ (f^{2} log \frac{f^{2}}{μf^{2}} ) ≦ 2cμ(f(−Lf))  ((LS))

with the same constant c ∈ (0, ∞) as before. Since it involves the Dirichlet form of the generator on its right hand side, once proven for L , it remains true, (possibly with a different constant), for any other Markov generator which has equivalent or larger Dirichlet form. We remark also that (LS) inequality has a tensorisation property which makes it suitable for infinite dimensional analysis.

We begin from describing the fundamentals of the theory of hypercontractive semigroups. It is followed by a description of the strong ergodicity strategy based on application of hypercontractivity. We conclude with presentation of numerous relations of hypercontractivity to spectral gap property, mixing and analyticity properties encountered in some applications. For a comprehensive description of the results and methods related to this lecture we refer to [23] .

Hypercontractivity and Logarithmic Sobolev Inequality.

Let P_{t} = e^{tL} be a Markov semi-group with an invariant measure μ . A square of the field Γ associated to the generator L is defined by

Γ(f, f) ≡ \frac{1}{2}(Lf^{2} − 2fLf)

One notes that for any P_{t} invariant measure μ

μΓ(f, f) = −μ(f(Lf))

Given p ∈ (1, ∞) and a constant c ∈ (0, ∞), we define

q(t) := q(t, p, c) = 1 + (p − 1) e^{\frac{2t}{c}} , t ∈ ℝ^{+} .

A key result in the theory of hypercontractive semigroup can be stated , (in a slightly more general form than indicated in the introduction), as follows.

Theorem (Gross Integration Lemma)

Let c ∈ (0, ∞) and d ∈ (0, ∞). The following statements are equivalent :

(i) For all p ∈ (1, ∞) and p ≤ q ≤ q(t, p, c), we have

||P_{t}f||_{q} ≤ exp(2d(\frac{1}{p} − \frac{1}{q})) ||f||_{p}  (73)

(ii) For p = 2 , q = 4 and all t ∈ R^{+},

||P_{t}f||_{4} ≤ e^{\frac{d}{2}} · ||f||_{2}  (74)

(iii)

μ(f^{2} log \frac{f}{||f||_{2}} ) ≤ cμΓ(f, f) + d||f||_{2}^{2}  (75)

for all non negative functions f for which the right hand side is finite. (iv) For some q ∈ [2, ∞),

μ(f^{q} log \frac{f}{||f||_{q}}) ≤ \frac{2c}{q}μΓ(f^{\frac{q}{2}} , f^{\frac{q}{2}} ) + \frac{2d}{q}||f||_{q}^{q}  (76)

for all non-negative functions for which the right hand side is finite. Properties (i)-(iv) imply

(v) For all t ∈ ℝ^{+} and any non negative function f such that ||f||_{1} = 1 we have

S_{f}(t) ≡ μf_{t} log f_{t} ≤ e^{\frac{−2t}{c}} S_{f}(0) + 2d(1 − e^{\frac{−2t}{c}})  (77)

for all t ∈ ℝ^{+}.

Remarks : We recall that the property (i) (and equivalently (ii)) with d = 0 is called (strict) hypercontractivity whereas the inequality (75) is called the logarithmic Sobolev inequality. The property (i) says that the semigroup P_{t} improve the smoothness by making any functions integrable in pth power also integrable with the some higher power. Moreover, the function t ⟶ exp(\frac{2d}{q(t)}) · ||P_{t}f||_{q(t)} is monotone nonincreasing and the idea of the proof of equivalence between (i) and (iv) is to show that it is differentiable and its derivative has appropriate properties iff (iv) is true. The Logarithmic Sobolev inequality for q = 1 can be interpreted as a bound of the relative entropy in terms of the Dirichlet form. In case of strict hypercontractivity we get from (v) the exponential decay to equilibrium in the entropy sense. (We remark that in general if additionally

μ ((Lf) log f) = −4μΓ(f^{\frac{1}{2}} , f^{\frac{1}{2}} )

then (v) is equivalent to (i)-(iv). This in particular holds if Γ is given by a proper gradient operator which satisfies the Leibnitz rule of differentiation of products of the functions. ) Equivalence of (iii) and (iv) follows from the appropriate covariance properties of the relative entropy and the L_{p} norms. Later on the constants (c, d), c ∈ (0, ∞) and d ∈ [0, ∞), such that (iii) is satisfied will be called the log-Sobolev coefficients.

Proof : The equivalence between (iii) and (iv) is easily obtained by substitution (For (iii)→(iv), we take f = g^{\frac{q}{2}} ; conversely (iv) can be clearly reduced to (iii) by choosing f = g^{\frac{2}{q}}).

To show that (i) implies (iv), we set f_{t} ≡ P_{t}f and consider the following derivative

∂_{t} log ||f_{t}||_{q(t)} = −\frac{∂_{t}q(t)}{q(t)^{2}} log μf_{t}^{q(t)} + \frac{1}{q(t)μf_{t}^{q(t)}} μ (f_{t}^{q(t)}∂_{t} (q(t) log f_{t}))
= −\frac{∂_{t}q(t)}{q(t)^{2}} log μf_{t}^{q(t)} + \frac{1}{q(t)μf_{t}^{q(t)}} μ((∂_{t}q(t))f_{t}^{q(t)} log f_{t}
+q(t)f_{t}^{q(t)−1}Lf_{t})  (78)
= \frac{∂_{t}q(t)}{q(t)μf_{t}^{q(t)}} (μ(f_{t}^{q(t)} log \frac{f_{t}}{||f_{t}||_{q(t)}} ) + \frac{cq(t)}{2(q(t) − 1)}μ(f_{t}^{q(t)−1}Lf_{t}))

where in the last line we used

∂_{t}q(t) = \frac{2}{c}(q(t) − 1).

which follows from our definition of q(t). To relate the factor in the bracket on the right hand side of (78) to (iv) we use the following special property (which follows from a representation of the classical Dirichlet form based on the integral representation P_{τ}f(x) = ∫ P_{τ}(x, dx̃) f(x̃)of the (symmetric) semigroup P_{τ} with a probability kernel P_{τ}(x, dx̃)) :

For any q ≥ 1 and any non-negative function g, one has

μ(g^{q−1}Lg) = lim_{τ→0} \frac{1}{τ}μ(g^{q−1}(P_{τ} − 1)g)
= − lim_{τ→0} \frac{1}{2τ} ∫ μ(dx)P_{τ}(x, dx̃)(g^{q−1}(x) − g^{q−1}(x̃))(g(x) − g(x̃))
≤ −\frac{4(q − 1)}{q^{2}} lim_{τ→0} \frac{1}{2τ} ∫ μ(dx)P_{τ}(x, dx̃)(g^{\frac{q}{2}} (x) − g^{\frac{q}{2}} (x̃))^{2} (79)
= −\frac{4(q − 1)}{q^{2}}μΓ(g^{\frac{q}{2}} , g^{\frac{q}{2}} ).

In (79) one uses the following inequality satisfied for any non negative numbers u, w,

(u^{\frac{q}{2}} − w^{\frac{q}{2}} )^{2} ≤ \frac{q^{2}}{4(q − 1)}(u^{q} − w^{q})(u − w).  (80)

The inequalities (78) and (79) yield

∂_{t} log ||f_{t}||_{q(t)}
≤ \frac{∂_{t}q(t)}{q(t)μf_{t}^{q(t)}} (μ(f_{t}^{q(t)} log \frac{f_{t}}{||f_{t}||_{q(t)}} ) − 2\frac{c}{q(t)}μΓ(ft^{\frac{q(t)}{2}} , ft^{\frac{q(t)}{2}} ))

Under hypothesis (iv), we deduce that

∂_{t} log ||f_{t}||_{q(t)} ≤ 2d\frac{∂_{t}q(t)}{q^{2}(t)}

and hence, taking into the account that q(0) = p, we arrive at the desired bound

||f_{t}||_{q(t)} ≤ ||f||_{p} exp\{2d(\frac{1}{p} − \frac{1}{q(t)})\}.

Finally, for any given q ≤ q(t), we can find a t_{0} ≤ t such that q = q(t_{0}). Using the definition of the semi-group and its contractivity property in any L_{p}, p ≥ 1, we obtain

||f_{t}||_{q} = ||f_{t}||_{q(t_{0})} = ||P_{t_{0}}f_{t−t_{0}}||_{q(t_{0})} ≤ ||f_{t−t_{0}}||_{p} exp\{2d(\frac{1}{p} − \frac{1}{q})\}
≤ ||f||_{p} exp\{2d(\frac{1}{p} − \frac{1}{q})\}

which extends the property to any q ≤ q(t).

To show that (i) implies (iv), we note first that (i) implies that a function

φ_{f}(t) : t→ exp\{\frac{2d}{q(t)}\}||f_{t}||_{q(t)} ,

defined for any non negative function f , is non-increasing. Indeed, (i) implies that for all s, t ≥ 0,

φ_{f}(t + s) ≤ exp\{\frac{2d}{q(t + s)}\}||P_{t}||_{q(s),q(t+s)}||f||_{q(s)}
≤ exp\{\frac{2d}{q(s)}\}||f||_{q(s)} = φ_{f}(s).

Consequently, log φ_{f}(t) is non-increasing and differentiable. Hence we obtain

∂_{t} log ||f_{t}||_{q(t)} ≤ ∂_{t}\{\frac{2d}{q(t)}\} = 2d\frac{∂_{t}q}{q^{2}} (t).

Going back to (78), we deduce

μ(f_{t}^{q} log \frac{f_{t}}{||f_{t}||_{q}}) + \frac{cq}{2(q − 1)}μ(f_{t}^{q−1}Lf_{t}) ≤ 2d\frac{μf_{t}^{q}}{q} (t).

Taking t = 0 and p = 2, we have

μ(f^{2} log \frac{f}{||f_{t}||_{2}} ) ≤ cμΓ(f, f) + d||f||_{2}^{2}.

Finally, to show that (iv) implies (v), let us differentiate with respect to the time variable the function

ψ_{f}(t) = e^{\frac{2t}{c}} S_{f}(t)

defined with a non negative f belonging to the domain of the generator L. Assuming that ||f||_{1} = 1 , (and so also ||f_{t}||_{1} = 1), we find that

∂_{t}e^{\frac{2t}{c}} S_{f}(t) = \frac{2}{c}e^{\frac{2t}{c}} S_{f}(t) + e^{\frac{2t}{c}} μ(Lf_{t} log f_{t}).  (81)

To estimate the last term, we note that for any nonnegative function F ∈ D(L),

μ[(LF) log F] = lim_{τ→0} \frac{−1}{2τ} ∫ μ(dx)P_{τ}(x, dx̃)(F (x) − F (x̃))(log F (x) − log F (x̃)).

Since for any non negative u, w one has the following limiting inequality (80),

(log u − log w)(u − w) ≥ (u^{\frac{1}{2}} − w^{\frac{1}{2}} )^{2},

we get

μ[(LF) log F] ≤ −4μΓ(F^{\frac{1}{2}} , F^{\frac{1}{2}} ).  (82)

From (81) and (82) we deduce that

∂_{t}e^{\frac{2t}{c}} S_{f}(t) ≤ \frac{2}{c}e^{\frac{2t}{c}} (S_{f}(t) − 2cμ(Γ(ft^{\frac{1}{2}} , ft^{\frac{1}{2}} ))

and, if logarithmic Sobolev inequality is satisfied,

∂_{t}e^{\frac{2t}{c}} S_{f}(t) ≤ \frac{4d}{c}e^{\frac{2t}{c}} .

Now (v) follows by integrating this differential inequality with respect to the time variable.

⋄

Properties of the Logarithmic Sobolev Inequality

Theorem (Product Property) Let (μ_{i})_{i=1,2} be two probability measures on probability spaces satisfying the logarithmic Sobolev inequality with coefficients (c_{i}, d_{i})_{i=1,2} for the infinitesimal generators (L_{i}, i = 1, 2). Let L ≡ L_{1} ⊗ 1 + 1 ⊗ L_{2} be a generator in the space C( 𝕄_{1} × 𝕄_{2}) with the corresponding square of the field form Γ ≡ Γ_{1}+Γ_{2}. Then, the product μ_{1} ⊗ μ_{2} satisfies

μ_{1} ⊗ μ_{2}(f^{2} log \frac{f}{||f||_{2}} ) ≤ cμ_{1} ⊗ μ_{2}(Γ(f, f) + dμ_{1} ⊗ μ_{2}f^{2}  (83)
with  c = max(c_{1}, c_{2})
d = d_{1} + d_{2}

for any non negative function f for which the right hand side of (83) is finite. Hence, if μ is a probability measure satisfying a logarithmic Sobolev inequality with a coefficient c < ∞ and d = 0, then the product μ^{⊗n} satisfies the logarithmic Sobolev inequality with the same coefficient c for any natural number n.

Proof : If f ∈ C( 𝕄_{1} ×𝕄_{2}) is in the domain of L , then, using the logarithmic Sobolev inequality for the μ_{1}, we obtain

μ_{1} ⊗ μ_{2}(f^{2} log \frac{f^{2}}{||f||_{2}^{2}} ) = μ_{2} (μ_{1}(f^{2} log \frac{f^{2}}{μ_{1}(f^{2})}) + μ_{1}(f^{2}) log \frac{μ_{1}(f^{2})}{μ_{1}⊗μ_{2}(f^{2})})
≤ μ_{2} (2c_{1}μ_{1}Γ_{1}(f, f) + 2d_{1}μ_{1}(f^{2}) + μ_{1}(f^{2}) log \frac{μ_{1}(f^{2})}{μ_{1}⊗μ_{2}(f^{2})})

To bound expectation of the second term we apply the logarithmic Sobolev inequality to a function g = \sqrt{μ_{1}(f^{2})} to get

μ_{1} ⊗ μ_{2}(f^{2} log \frac{f^{2}}{||f||_{2}^{2}} ) ≤ 2c_{1}μ_{2}⊗μ_{1}Γ_{1}(f, f) + 2c_{2}μ_{2}Γ_{2}(g, g)
+2(d_{1} + d_{2})μ_{2}⊗μ_{1}(f^{2}).

The last term of the above inequality is bounded as follows

μ_{2}Γ_{2}(g, g) = μ_{2}Γ_{2}(\sqrt{μ_{1}(f^{2})}, \sqrt{μ_{1}(f^{2})})
= lim_{τ↓0} \frac{1}{2τ} ∫ μ_{2}(dx)P_{τ,2}(x, dx̃) (\sqrt{μ_{1}(f^{2})}(x) − \sqrt{μ_{1}(f^{2})}(x̃))^{2}
≤ lim_{τ↓0} \frac{1}{2τ} ∫ μ_{2}(dx)P_{τ,2}(x, dx̃)dμ_{1}(dη) (f(x, η) − f(x̃, η))^{2}
= μ_{1} ⊗ μ_{2} (Γ_{2}(f, f))

where P_{τ,2}(x, dx̃) denotes the kernel of the semigroup generated by 1 ⊗ L_{2}, and the above inequality follows from

|\sqrt{μ_{1}(f^{2})}(x) − \sqrt{μ_{1}(f^{2})}(x̃)| = | ||f(x, ·)||_{L_{2}(μ_{1})} − ||f(x̃, ·)||_{L_{2}(μ_{1})} |
≦ ||f(x, ·) − f(x̃, ·)||_{L_{2}(μ_{1})}

(which is a consequence of the Minkowski inequality). Hence

μ_{1} ⊗ μ_{2}(f^{2} log \frac{f^{2}}{||f||_{2}^{2}} )

≤ 2c_{1}μ_{2}⊗μ_{1}Γ_{1}(f, f) + 2c_{2}μ_{2} ⊗ μ_{1}(Γ_{2}(f, f)) + 2(d_{1} + d_{2})μ_{2}⊗μ_{1}(f^{2})
≤ 2 max(c_{1}, c_{2})μ_{2}⊗μ_{1}Γ(f, f) + 2(d_{1} + d_{2})μ_{2}⊗μ_{1}(f^{2})

which ends the proof.

⋄

Theorem (Spectral Gap) If μ is a probability measure satisfying the Logarithmic Sobolev inequality with coefficients (c, 0), then μ satisfies the following spectral gap inequality

mμ(f − μf)^{2} ≦ μΓ(f, f)

with a coefficient

m ≥ \frac{1}{c}.

Conversely, if μ satisfies logarithmic Sobolev inequality with coefficients (c, d) as well as a spectral gap inequality with coefficient m, then μ satisfies logarithmic Sobolev inequality with coefficients (c′, 0) with

c′ ≤ c + \frac{d + 1}{m}.

Proof : To prove the spectral gap property we consider a bounded function

f ∈ D(L) which has mean value equal to zero. Than for ε sufficiently small, 1 + εf is a positive bounded function in the domain of L. Hence, applying logarithmic Sobolev inequality, we obtain

μ(1 + εf)^{2} log(1 + εf)^{2} ≤ 2cε^{2}μ(Γ(f, f)) + μ(1 + εf)^{2} log μ(1 + εf)^{2}.(84)

By Taylor’s expansion , we get

μ(1 + εf)^{2} log(1 + εf)^{2} = 3ε^{2}μf^{2} + O(ε^{3})

and

μ(1 + εf)^{2} log μ(1 + εf)^{2} = ε^{2}μf^{2} + O(ε^{3})

Inserting this to the inequality (84), dividing both sides by ε^{2} and passing with ε to zero, we arrive at

μf^{2} ≤ cμ(Γ(f, f)).

To prove the second point of the theorem, it is sufficient to show that for any measurable bounded function f, one has

μ(f^{2} log \frac{f^{2}}{||f||_{2}^{2}} ) ≤ μ((f − μf)^{2} log \frac{|f − μf|^{2}}{||f − μf||_{2}^{2}} ) + 2||f − μf||_{2}^{2}. (85)

Since then the desired statement is obtained by using the logarithmic Sobolev inequality and the spectral gap inequality to bound the first and the second term on the right hand side (85), respectively. To show (85), note first that it is equivalent, for μf ≠ 0 to

μ((1 + tψ)^{2} log \frac{(1 + tψ)^{2}}{1 + t^{2}} ) ≤ t^{2}μψ^{2} log ψ^{2} + 2t^{2}  (86)

with

t = \frac{||f − μf||_{2}}{μf} ψ = \frac{f − μf}{||f − μf||_{2}} .

We shall use the following observations concerning ψ ;

μψ = 0, μ(ψ^{2}) = 1.  (87)

Changing ψ into −ψ, we can assume that t ≥ 0. To finish the proof it is sufficient to study a function

F (t) = μ((1 + tψ)^{2} log \frac{(1 + tψ)^{2}}{1 + t^{2}} )

and show that its second derivative is bounded by an appropriate constant. Using (87), we find

F′(t) = μ(2ψ(1 + tψ) log \frac{(1 + tψ)^{2}}{1 + t^{2}} )

and then

F′′(t) = μ(2ψ^{2} log \frac{(1 + tψ)^{2}}{1 + t^{2}} ) + 4 − \frac{4t}{1 + t^{2}} .

We bound the first term of F′′(t) by using Jensen’s inequality applied to the probability measure μ(ψ^{2} · ) as follows

μ(ψ^{2} log \frac{(1 + tψ)^{2}}{1 + t^{2}} ) − μ(ψ^{2} log ψ^{2}) = μ (ψ^{2} (log \frac{(1 + tψ)^{2}}{ψ^{2}(1 + t^{2})})) ≦
≦ log μ (ψ^{2} \frac{(1 + tψ)^{2}}{ψ^{2}(1 + t^{2})}) = 0,

Hence , for any t ≥ 0 we have

F′′(t) ≤ 2μψ^{2} log ψ^{2} + 4.

Integrating this inequality twice with the initial conditions F′(0) = F (0) = 0 ends the proof of the necessary result (86). Note that in general F may not be differentiable in which case we need to apply the same arguments to an auxiliary function

F_{δ}(t) = μ([δ^{2} + (1 + tψ)^{2}] log \frac{[δ^{2} + (1 + tψ)^{2}]}{1 + t^{2} + δ^{2}})

defined with some non negative real number δ. One can then show that

F′′_{δ}(t) ≤ 2μψ^{2} log ψ^{2} + 4(1 + δ^{2})

and conclude as above with further use of dominated convergence theorem.

Exercise

• Use (ii) of the integration lemma together with both parts on spectral gap theorem to prove the logarithmic Sobolev inequality for a measure μ = pδ_{+1} + (1 − p)δ_{−1} on two point set.

• Prove the logarithmic Sobolev inequality for the two point measure by a direct computation, (see [21] for the uniform case).

• Use the result from the exercises above together with the product property of (LS) and the central limit theorem to show that (LS) is true for a Gaussian measure.

⋄

Strong Ergodicity via Hypercontractivity

In this section we describe how one can use the hypercontractivity to prove the strong ergodicity of Markov semigroups in infinite dimensions.

Let 𝕄 be a finite set or a compact connected and smooth Riemannian manifold. Let {μ_{X}^{x} : X ⊂⊂ ℤ^{d}, x ∈ 𝕄^{ℤ^{d}}} be a local specification defined with a uniform product measure μ_{0} = ν^{⊗ℤ^{d}} , given by some probability measure ν on 𝕄 , and a finite range interaction potential Φ as follows

μ_{X}^{x}(f) ≡ \frac{∫ ν^{⊗X}(dy_{X})e^{−U_{X}(y_{X}·x_{ℤ^{d}\X})}f(y_{X} · x_{ℤ^{d}\X})}{∫ ν^{⊗X}(dy_{X})e^{−U_{X}(y_{X}·x_{ℤ^{d}\X})}}

where

(y_{X} · x_{ℤ^{d}\X})_{j} = \{ \begin{matrix}y_{j} & if & j ∈ X \\ x_{j} & if & j ∈ ℤ^{d} \ X \end{matrix}

We recall that a probability measure μ on 𝕄^{ℤ^{d}} is called a Gibbs measure for the local specification iff μ satisfies the following Dobrushin-Lanford-Ruelle equation

μ (μ_{X}^{·}(f)) = μ(f)  ((DLR))

for all bounded measurable functions. In our setting one has the following key result saying that under the assumption of (strict) hypercontractivity we have the decay to equilibrium in uniform norm with essentially the same exponential rate as in L_{2}(μ) .

Theorem Let μ be a Gibbs measure on 𝕄^{ℤ^{d}} which satisfies the Logarithmic Sobolev Inequality with a coefficient c ∈ (0, ∞) (and d = 0) .

Suppose that there exists a finite constant C_{1} such that for any non negative function f, any Λ ⊂⊂ ℤ^{d},

||P_{t=1}^{(Λ)}f||_{∞} ≤ e^{C_{1}|Λ|}μ_{Λ}^{·}(f).  (88)

Moreover, assume that the following exponential approximation property is satisfied : any A ∈ ℝ^{+}

||P_{t}f − P_{t}^{(Λ)}f||_{∞} ≤ e^{−At}|||f|||  (89)

for any local function f provided that for some sufficiently large L,

dist(Λ(f), Λ^{c}) ≥ Bt

with a finite constant B > 0 dependent only on A, whenever dist(Λ(f), Λ^{c}) ≧ L.

Then, for any θ ∈ (0, 1), there exist a finite constant C(θ, Λ(f)) and a positive real number m ∈ [1/c, ∞) such that for any t ≥ 0,

||P_{t}f − μf||_{∞} ≤ C(θ, Λ(f))e^{−θmt}|||f|||.

Remark : The property (88), called ultracontractivity, is trivially satisfied if 𝕄 is a finite set. Under some additional conditions frequently encountered in applications, it also holds true in case of continuous finite dimensional configuration space.

As discussed briefly in Lecture I, the approximation property (89) follows from the construction of the dissipative dynamics.

Before going into the technicalities, we can describe the idea of the proof as follows. First one needs to use the approximation property (89) to pass to the finite dimensions in which we can apply the ultracontractive bound (using only a unit of time). This brings us to the analysis in L_{p} spaces, but there is a huge prise we pay for that in the form of a factor exponentially growing with respect to the dimension. Fortunately we can now apply hypercontractivity for a small fraction of time to neutralise that factor. Additionally this route leads us to the Hilbert space L_{2} and the rest follows from the spectral analysis, (since by general arguments we know that Logarithmic Sobolev inequality (with d = 0) implies the spectral gap between zero eigenvalue and the rest of the spectrum of the generator).

Proof : We note first that

|P_{t}f − μf| ≤ |P_{t}^{(Λ)}f − μf| + |P_{t}^{(Λ)}f − P_{t}f|.  (90)

The second term will be estimated by the exponential approximation property. For the first term on the right hand side of (90), we note that, by Hölder’s inequality, for any q ≥ 1,

|P_{t}^{(Λ)}f − μf| = |P_{1}^{(Λ)}(P_{t−1}^{(Λ)}f − μf)|
≤ |P_{1}^{(Λ)}|P_{t−1}^{(Λ)}f − μf|^{q}|^{\frac{1}{q}}
≤ e^{\frac{C_{1}|Λ|}{q}} (μ_{Λ}^{·}|P_{t−1}^{(Λ)}f − μf|^{q})^{\frac{1}{q}}

where in the last line we used hypothesis (88). Moreover, by definition of P_{t}^{(Λ)}, if Λ(f) ⊂ Λ , then for any time t, P_{t}^{(Λ)}f − μf depends only on x_{Λ^{R}} , where Λ^{R} ≡ {i : d(i, Λ) ≤ R}. Hence, (using the assumption that the potential Φ is bounded and of finite range), we can find a finite constant C_{2} so that

μ_{Λ}^{·}|P_{t−1}^{(Λ)}f − μf|^{q} ≤ e^{C_{2}|∂Λ|}μ|P_{t−1}^{(Λ)}f − μf|^{q}.

Noting as well that

(μ|P_{t−1}^{(Λ)}f − μf|^{q})^{\frac{1}{q}} ≤ ||P_{t−1}^{(Λ)}f − P_{t−1}f||_{∞} + (μ|P_{t−1}f − μf|^{q})^{\frac{1}{q}} (91)

we deduce from (90) - (91) that

||P_{t}^{(Λ)}f − μf||_{∞} ≤ e^{\frac{C_{1}|Λ|+C_{2}|∂Λ|}{q}} ((μ(P_{t−1}f − μf)^{q})^{\frac{1}{q}} + ||P_{t}^{(Λ)}f − P_{t}f||_{∞}) . (92)

To estimate the right hand side of (92) we will use the finite volume approximation property. To this end we chose Λ sufficiently large, so that Λ(f) ⊂ Λ and its diameter diam(Λ) ∼ t ≥ 1 ( and so |Λ| ∼ t^{d} ). With this choice we obtain the following bound

||P_{t}^{(Λ)}f − μf||_{∞} ≤ e^{\frac{C′_{1}t^{d}}{q}} (μ(P_{t−1}f − μf)^{q})^{\frac{1}{q}} + e^{\frac{C′_{1}t^{d}}{q}} e^{−At}|||f|||

for a finite constant C′_{1}. If moreover μ satisfies the Logarithmic Sobolev inequality with a coefficient c < ∞, choosing q ≦ q(\frac{t−1}{θt}, 2, c) = 1 + e^{\frac{2(1−θ)t−2}{c}} with some θ ∈ (0, 1) , by the Gross integration lemma we arrive at

(μ|P_{t−1}f − μf|^{q})^{\frac{1}{q}} ≤ (μ(P_{θt}f − μf)^{2})^{\frac{1}{2}} ≤ e^{−mθt}||f − μf||_{2}

where in the last inequality we have used the spectral gap property for the generator L , (discussed in the previous section), which allows us to take

m ≧ c^{−1} . With such the choice of q, the factor e^{\frac{C′_{1}t^{d}}{q}} is uniformly bounded with respect to t and we obtain the desired estimate.

⋄

Exercise : Show that by using the generalised approximation property, (as described in the exercise in Lecture I), one can extend the above theorem to the case of long range potentials in Φ ∈ B_{2}.

Equivalence Theorem

We consider a local specification {μ_{Λ}^{x} : Λ ⊂⊂ ℤ^{d}, x ∈ 𝕄^{ℤ^{d}}} defined by a product measure μ_{0} = ν^{⊗ℤ^{d}} and an interaction potential Φ = (Φ_{X})_{X⊂⊂ℤ^{d}} with finite range R. If necessary, we shall indicate more explicitly the potential defining the local specification by using the following notation {μ_{Λ,Φ}^{x} : Λ ⊂⊂ ℤ^{d}, x ∈ 𝕄^{ℤ^{d}}}.

For Λ ⊂⊂ ℤ^{d}, let P_{t}^{(Λ,x)} = e^{tL_{Λ,x}} be the Markov semi-group generated by

L_{Λ,x} = δ_{x_{Λ^{c}}}(∑_{i∈Λ} L_{i})

that is the semigroup acting on the variables x_{Λ} ≡ {x_{i}, i ∈ Λ} and with fixed boundary conditions, where we used the notation

(δ_{x_{Λ^{c}}}L_{i})f(x̃) = μ_{{i}}f(x̃_{Λ∩{i}^{c}} ∘ x_{Λ^{c}∩{i}^{c}}) − f(x̃_{Λ} ∘ x_{Λ^{c}})

The following result of [39] relates a number of conditions describing different features of the system.

Theorem The following conditions are equivalent (i) \underline{Strong mixing conditions} There exists ε ∈ (0, ∞) such that for any Λ ⊂⊂ ℤ^{d} , any x ∈ 𝕄^{ℤ^{d}}, and

any local functions f and g localized in Λ,

|μ_{Λ}^{x} ((f − μ_{Λ}^{x}(f))g) | ≤ e^{−εd(Λ(f),Λ(g))}|||f|||.|||g|||.

(ii)\underline{Complete analyticity condition :}

For any n ∈ ℕ and any potentials {ψ_{k}, k = 1, .., n} with finite range, the map

(z_{1}, .., z_{n}) ⟶ μ_{Λ, Φ+∑_{k=1}^{n} z_{k}ψ_{k}}^{x} (f)

is analytic in a neighborhood ∩{|z_{k}| ≤ δ} of the origin for some δ independent of Λ ⊂⊂ ℤ^{d}, x ∈ 𝕄^{ℤ^{d}} and of the local function f .

(iii)\underline{Uniform spectral gap inequality :}

There exists a constant m ∈ (0, ∞) such that for any Λ ⊂⊂ ℤ^{d}, and all x ∈ 𝕄^{ℤ^{d}},

m μ_{Λ}^{x}(f − μ_{Λ}^{x}f)^{2} ≤ μ_{Λ}^{x}(|∇f|^{2})

for any function f for which the right hand side of the above inequality is well defined and finite.

(iv)\underline{Uniform logarithmic Sobolev inequality :}

There exists a constant c ∈ (0, ∞) such that for any Λ ⊂⊂ ℤ^{d}, and all x ∈ 𝕄^{ℤ^{d}},

μ_{Λ}^{x}(f log \frac{f}{μ_{Λ}^{x}f}) ≤ c μ_{Λ}^{x}(|∇f|^{2})

for any non negative function f for which the right hand side of the above inequality is well defined and finite.

Remarks : For detailed discussion we refer to [23] and references therein.

(1) The equivalence between (i) and (ii) is due to Dobrushin and Shlosman [17] who proved the equivalence between some 14 conditions encountered in equilibrium statistical mechanics. The proof of the equivalence with (iii) and (iv) was given first in [39]. ( Similar conditions remain equivalent if one replaces the sequence of finite sets which are unions of cubes of a given sufficiently large size. In this setting, the equivalence between (i) and (ii) is due to E. Olivieri and P. Picco, the others have been considered by F.Martinelli and E. Olivieri and S.Lu and H.T. Yau. These results allow to extend the domain of validity of logarithmic Sobolev inequalities to a larger class of potentials.)

(2) The implication of (iii) given (iv) was shown before as a general direct consequence of the Logarithmic Sobolev Inequality. The inverse implication is rather unusual. It is essentially related to the assumption that the spectral gap inequality is satisfied uniformly with respect to the finite sets Λ and the boundary conditions x.

(3) For details of the proof of (iv) assuming (i) see [23] (and references therein).

(4) The fact that (iii) implies (i) is due to the exponential approximation property. Indeed, we have

μ_{Λ}^{x}((f − μ_{Λ}^{x}(f))(g − μ_{Λ}^{x}(g)) = μ_{Λ}^{x}(P_{t}^{(Λ,x)}(fg)) − μ_{Λ}^{x}(f)μ_{Λ}^{x}(g)

Using the exponential approximation property, we get

|P_{t}^{(Λ,x)}(fg) − P_{t}^{(Λ,x)}fP_{t}^{(Λ,x)}g| ≤ e^{−At}|||f|||.|||g|||

provided that

d(Λ(f), Λ(g)) ≈ Bt

with some B = B(A) ∈ (0, ∞). As a consequence,

|μ_{Λ}^{x}(f, g)| ≤ |μ_{Λ}^{x}(P_{t}^{(Λ,x)}f, P_{t}^{(Λ,x)}g)| + e^{−At}|||f|||.|||g|||
≤ (μ_{Λ}^{x}(P_{t}^{(Λ,x)}f − μ_{Λ}^{x}f)^{2}μ_{Λ}^{x}(P_{t}^{(Λ,x)}g − μ_{Λ}^{x}g)^{2})^{\frac{1}{2}} + e^{−At}|||f|||.|||g|||

Using the spectral gap inequality, we obtain (i).

Lecture III

Logarithmic Sobolev Inequality in

Non-Commutative Setting

1. Introduction

In this lecture we are setting a general scene for hypercontractivity and its infinitesimal description in the noncommutative spaces. To have possibly explicit and useful for intuition exposition we choose to work with an interpolating family of 𝕃_{p} spaces associated to a faithful normal state given by a normalised strictly positive density matrix with respect to a normalised trace. As one could notice in the previous lecture we need to consider carefully geometric and analytic features of the chosen interpolating family of Banach spaces, including relations of positivity in algebra an in the spaces, isometric embeddings and differentiability property of the norms related to the natural flow related to them. It is this last property which leads us to an intrinsic notion of entropy and its corresponding operator valued counterpart. Our construction assures that it has the appropriate covariance property (required for logarithmic Sobolev inequality). In this framework we recover basically all essential properties of the hypercontractivity theory and the related infinitesimal condition. We remark that the sufficiency proof of the integration lemma is essentially based on suitable properties of the corresponding 𝕃_{p}- Dirichlet forms, which in commutative setting , (or in case when the state is given by a trace), follows easily from the representation theorem for Dirichlet forms, but may be not obvious in noncommutative spaces. It turns out also that the tensorisation property of logarithmic Sobolev inequality appears to be difficult to prove directly, (though we essentially recover it in the next lecture by a different method).

We begin from describing the basic properties of the underlying 𝕃_{p} spaces which is followed by a general description of Markov semigroups in this setting. The key part concerning hypercontractivity and its infinitesimal description is contained in section 4.

The main reference for this lecture is [36] .

2. Noncommutative 𝕃_{p} spaces.

Let ω( · ) = Tr(ρ · ), with a strictly positive density matrix ρ ∈ A, with respect to a normalized trace Tr and satisfying Trρ = 1. For each p ∈ [1, ∞) we are given the following norm

∥f∥_{𝕃_{p}(ω)} = (Tr|ρ^{\frac{1}{2p}} fρ^{\frac{1}{2p}} |^{p})^{1/p}

where |a| ≡ (a^{∗}a)^{\frac{1}{2}} . The corresponding 𝕃_{p}(ω) space for p ∈ [1, ∞) is defined as the closure of A in this norm. 𝕃_{∞}(ω) is defined as the dual space to 𝕃_{1}(ω). Such definition assures that we have:

A ⊂𝕃_{∞}(ω) ⊂𝕃_{p}(ω) ⊂ 𝕃_{q}(ω) ⊂ 𝕃_{1}(ω)

if p ≥ q and algebra A is dense in each 𝕃_{p}(ω). It is also clear that 𝕃_{2}(ω) is a Hilbert space with the following scalar product

< f, g >_{ω} = Tr(ρ^{\frac{1}{4}} fρ^{\frac{1}{4}} )^{∗}(ρ^{\frac{1}{4}} gρ^{\frac{1}{4}} )

Exercise :

• To see that the corresponding 𝕃_{p}(ω) can be defined for more general states, note the following : For p = 2 one has

∥f∥_{𝕃_{2}(ω)}^{2} = Tr|ρ^{\frac{1}{4}} fρ^{\frac{1}{4}} |^{2} = Tr (ρ^{\frac{1}{2}} f^{∗}ρ^{\frac{1}{2}} f)
= ω (f^{∗}(ρ^{\frac{1}{2}} fρ^{− \frac{1}{2}} )) ≡ ω(f^{∗}α_{\frac{i}{2}} (f))

where α_{t} is the corresponding modular automorphism. Find similar formula involving modular automorphism for norms with even integer p and show that they are related to n-tuple forms possessing appropriate linearity or antilinearity properties .

• Show that

< 1, f >_{ω}= ω(f)

Basic properties of these spaces are collected in the following theorem (see e.g.[48]):

Theorem 2.1.

(a) Hölder type inequalities :

• The scalar product < A, A >_{ω} can be extended to <𝕃_{p}(ω), 𝕃_{q}(ω) >_{ω}, with p, q ∈ [1, ∞] satisfying \frac{1}{p} + \frac{1}{q} = 1, and

| < f, g >_{ω} | ≤ ∥f∥ _{𝕃_{p}(ω)}∥g∥ _{𝕃_{q}(ω)}  (93)

• If p, q ∈ [1, ∞], p ≤ q , then

∥f∥ _{𝕃_{p}(ω)} ≤ ∥f∥ _{𝕃_{q}(ω)}  (94)

(b) Duality: For any p ∈ [1, ∞] , we have

∥f∥_{𝕃_{p}(ω)} = sup_{∥g∥ _{𝕃q(ω)}=1} | < g, f >_{ω} |  (95)

supremum being attained for the following (normalised) tangent vector defined for f ≠ 0 by

ϕ_{q}(f) ≡ ∥f∥_{𝕃_{p}(ω)}^{1−p}ρ^{− \frac{1}{2q}} |ρ^{\frac{1}{2p}} fρ^{\frac{1}{2p}} |^{p−2} (ρ^{\frac{1}{2p}} f^{∗}ρ^{\frac{1}{2p}} ) ρ^{− \frac{1}{2q}}

where , for p ∈ (1, ∞) , one has \frac{1}{p} + \frac{1}{q} = 1, and by definition ϕ_{q}(0) ≡ 0.

Remark : The family {𝕃_{p}(ω)}_{1≤p≤∞} is a (complex) interpolating family of Banach spaces, [48] .

Exercise : Prove Theorem 2.1 for ω = Tr . Show also that for p, q, r ∈ [1, ∞], \frac{1}{p} + \frac{1}{q} = \frac{1}{r} one has

∥fg∥ _{𝕃_{r}(Tr)} ≤ ∥f∥ _{𝕃_{p}(Tr)}∥g∥ _{𝕃_{q}(Tr)}  (96)

Hint : Note first that for a finite dimensional space H , by Cauchy-Schwartz inequality we have

| < F, G >_{Tr} | ≡ |\frac{1}{dim H} ∑_{i,j=1,..,dim H} \overline{F}_{ij}G_{ij}| ≦

\frac{1}{dim H} ∑_{i=1,..,dim H} ( ∑_{j=1,..,dim H} \overline{F}_{ij}F_{ij})^{\frac{1}{2}} ( ∑_{j=1,..,dim H} \overline{G}_{ij}G_{ij})^{\frac{1}{2}} = Tr |F | |G|

where |F | ≡ (F^{∗}F)^{\frac{1}{2}} and |G| ≡ (G^{∗}G)^{\frac{1}{2}} . Hence, it is enough to consider positive operators.

Proof : To prove the Hölder inequalities we use the corresponding inequalities for the trace as follows. Note first that if F and G are operators on finite dimensional Hilbert space and ω(·) = Tr(ρ·), then for \frac{1}{p} + \frac{1}{q} = 1 we have

< F , G >_{ω}= Tr (ρ^{\frac{1}{2}} F^{∗}ρ^{\frac{1}{2}} G) = Tr (ρ^{\frac{1}{2p}} F^{∗}ρ^{\frac{1}{2p}} ) (ρ^{\frac{1}{2q}} Gρ^{\frac{1}{2q}} )
≦ ||ρ^{\frac{1}{2p}} F ρ^{\frac{1}{2p}} || _{𝕃_{p}(Tr)}||ρ^{\frac{1}{2q}} Gρ^{\frac{1}{2q}} || _{𝕃_{q}(Tr)} = ||F || _{𝕃_{p}(ω)}||G|| _{𝕃_{q}(ω)}

To prove the second Hölder (94) inequality, we note that for p, q ∈ [1, ∞], p ≦ q , setting \frac{1}{2r} ≡ \frac{1}{2p} − \frac{1}{2q} we have

||F || _{𝕃_{p}(ω)} = ||ρ^{\frac{1}{2p}} F ρ^{\frac{1}{2p}} || _{𝕃_{p}(Tr)} = (Tr |ρ^{\frac{1}{2r}} (ρ^{\frac{1}{2q}} F ρ^{\frac{1}{2q}} ) ρ^{\frac{1}{2r}} |)^{\frac{1}{p}}  (97)
≦ ||ρ^{\frac{1}{2r}} || _{𝕃_{2r}(Tr)}|| (ρ^{\frac{1}{2q}} F ρ^{\frac{1}{2q}} ) ρ^{\frac{1}{2r}} || _{𝕃_{2s}(Tr)} = || (ρ^{\frac{1}{2q}} F ρ^{\frac{1}{2q}} ) ρ^{\frac{1}{2r}} || _{𝕃_{2s}(Tr)}
≦ ||ρ^{\frac{1}{2q}} F ρ^{\frac{1}{2q}} || _{𝕃_{2q}(Tr)}||ρ^{\frac{1}{2r}} || _{𝕃_{2r}(Tr)} = ||ρ^{\frac{1}{2q}} F ρ^{\frac{1}{2q}} || _{𝕃_{2q}(Tr)} = ||F || _{𝕃_{q}(ω)}

where we have used the Hölder inequality (96) for the trace twice, with \frac{1}{2r} ≡ \frac{1}{2p} − \frac{1}{2q} and \frac{1}{2r} + \frac{1}{2s} = \frac{1}{p}. The property (c) follows from (93), since by inspection

< ϕ_{q}(f), f >_{ω}= ∥f∥ _{𝕃_{p}(ω)}  (98)

This ends the proof of Theorem 2.1. □

For later purposes we introduce the cones of positive elements in𝕃_{p}(ω) , by introducing the 𝕃_{p} - modulus | · |_{p} , later on called briefly p-modulus, given for any f ∈ A as follows

|f|_{p} ≡ ρ^{− \frac{1}{2p}} |ρ^{\frac{1}{2p}} fρ^{\frac{1}{2p}} |ρ^{− \frac{1}{2p}}  (99)

Proposition 2.2. The p−modulus given on A by (99) extends by continuity to an isometry |·|_{p} on 𝕃_{p}(ω) and thus the mapping 𝕃_{p}(ω) ∋ f → |f|_{p} ∈ 𝕃_{p}(ω) is a continuous projection which satisfies

∥ |f|_{p}∥ _{𝕃_{p}(ω)} = ∥f∥ _{𝕃_{p}(ω)}

The proof follows by use of the following inequality for the trace norms [7]

∥ |F | − |G| ∥ _{𝕃_{p}(Tr)} ≤ 2^{1/2}(∥ |F | + |G| ∥ _{𝕃_{p}(Tr)})^{1/2}(∥ |F − G| ∥ _{𝕃_{p}(Tr)})^{1/2}

Definition : The cone of positive elements in 𝕃_{p}(ω) is defined by

𝕃_{p}^{+}(ω) = {f ∈ 𝕃_{p}(ω) : |f|_{p} = f}

when p ∈ [1, ∞) and

𝕃_{∞}^{+}(ω) = {f ∈ 𝕃_{∞}(ω) : < f, g >_{ω}≥ 0 ∀g ∈ 𝕃_{1}^{+}(ω)}

Remark For every p ∈ [1, ∞) , 𝕃_{p}^{+}(ω) is a closed strict cone in 𝕃_{p}(ω) such that A^{+} is dense in it. This remains true for p = ∞, provided the norm topology is replaced by the weak^{∗}-topology

The following proposition allows to construct appropriate decomposition into positive and negative parts in a space 𝕃_{p}^{R}(ω) of symmetric operators in 𝕃_{p}(ω).

Proposition 2.3. For any f ∈ 𝕃_{p}^{R}(ω), p ∈ [1, ∞), we have

|f|_{p} ± f ∈ 𝕃_{p}^{+}(ω)

Proof : For a ∈ A^{R} ≡ {b ∈ A : b^{∗} = b}, let

ρ^{\frac{1}{2p}} aρ^{\frac{1}{2p}} = A_{+} − A_{−}

be the decomposition of the hermitian operator ρ^{\frac{1}{2p}} aρ^{\frac{1}{2p}} with respect to the natural cone of positive elements in A. Then |ρ^{\frac{1}{2p}} aρ^{\frac{1}{2p}} | = A_{+} + A_{−} and so |a|_{p} = ρ^{− \frac{1}{2p}} (A_{+} + A_{−})ρ^{− \frac{1}{2p}}. It follows that

|a|_{p} ± a = 2ρ^{− \frac{1}{2p}} A_{±}ρ^{− \frac{1}{2p}} ∈ 𝕃_{p}^{+}(ω)

Since A^{R} is dense in 𝕃_{p}^{R}(ω) , so for some sequence a_{n} ∈ A we have

|f|_{p} ± f = lim_{n→∞}(|a_{n}|_{p} ± a_{n}) ∈𝕃_{p}^{+}(ω)

because 𝕃_{p}^{+}(ω) is closed. □

Remark : It follows from the construction that for \frac{1}{p} + \frac{1}{q} = 1, one has

< 𝕃_{p}^{+}(ω), 𝕃_{q}^{+}(ω) >_{ω}≥ 0

Now if q ∈ (1, ∞), by the duality property, for any f ∈ 𝕃_{q}^{+}(ω) we have the following representation of its norm

∥f∥_{𝕃_{q}(ω)} = sup{< g, f >_{ω} : g ∈ 𝕃_{p}^{+}(ω), ∥g∥_{𝕃_{p}(ω)} = 1}

with \frac{1}{p} + \frac{1}{q} = 1. This supremum is attainable.

3. Hypercontractive Semigroup in Noncommutative 𝕃_{p} spaces.

Let 𝕃_{p} and 𝕃_{p}^{+}, p ∈ [1, ∞], be an interpolating family of Banach spaces and its positive cone, which contain a Banach space A and A^{+} as a common dense subset, respectively, and such that the pairs with complementary indexes p, q ∈ (1, ∞), \frac{1}{p} + \frac{1}{q} = 1, are dual with respect to the scalar product < · , · > of the Hilbert space 𝕃_{2}.

Definition 3.1. A strongly continuous semigroup P_{t} of linear operators in the interpolating family 𝕃_{p} is called:

(i) Contractive iff for each p and any t ≥ 0, ||P_{t}f∥_{p} ≤ ∥f∥_{p}

(ii) Positive iff for each p and any t ≥ 0, P_{t}( 𝕃_{p}^{+} ) ⊂𝕃_{p}^{+}

(iii) 𝕃_{2}-symmetric iff < f, P_{t}g > = < P_{t}f, g > ∀f, g ∈𝕃_{2}

(iv) Markov iff it is positive, contractive and P_{t}(1) = 1, where 1 is the

unit operator in A

(v) Feller iff P_{t}A ⊂ A and P_{t}|_{A} is positive and contractive

(vi) Hypercontractive iff it is contractive and for every 1 < q < p < ∞

there exist τ_{p, q} ≥ 0 and C_{p, q} ∈ (0, ∞) such that ∀t > τ_{p, q} the norm ∥P_{t}∥_{p,q} of the operator P_{t} : 𝕃_{q} ⟶ 𝕃_{p} satisfies ∥P_{t}∥_{p,q} ≤ C_{p, q}

(vii) Strictly Hypercontractive iff the constant C_{p, q} ≤ 1.

Remarks Notice that a positive and contractive semigroup on 𝕃_{∞} with the property < f, P_{t}g > = < P_{t}f, g > for all f, g ∈𝕃_{∞} can be extended in every 𝕃_{p} space to a positive semigroup of contractions. Indeed for f ∈𝕃_{∞}, setting f_{t} = P_{t}f , we have

∥P_{t}f|| _{𝕃_{1}} = < ϕ_{∞}(f_{t}), P_{t}f > = < P_{t}ϕ_{∞}(f_{t}), f >≤
≤ ∥P_{t}ϕ_{∞}(f_{t})∥ _{𝕃_{∞}}∥f∥ _{𝕃_{1}} ≤ ∥ϕ_{∞}(f_{t})∥ _{𝕃_{∞}}∥f∥ _{𝕃_{1}} = ∥f∥ _{𝕃_{1}}

Hence ∥P_{t}f∥ _{𝕃_{1}} ≤ ∥f∥ _{𝕃_{1}}. The contractivity for arbitrary p ∈ (1, ∞) follows from the Riesz-Thorin interpolation theorem. Because 𝕃_{∞}^{+} is dense in each 𝕃_{p}^{+}, so the extension of P_{t} onto 𝕃_{p} is also positive. By continuity this extension is 𝕃_{2}-symmetric. We can show in a similar way that such an extension is also possible if given a positive semigroup of contractions in 𝕃_{1} which is Feller and satisfies < P_{t}f, g > = < f, P_{t}g > ∀f, g ∈ A .

Recall that one can give an infinitesimal characterization of the 𝕃_{p} - contractivity property in terms of the generator of a semigroup. Let L_{q} be the generator of the semigroup P_{t} in 𝕃_{q}, with the domain denoted by D(L_{q}). (If no confusion can arise we omit the subscript q from the notation.) Let E_{2}( ·, · ) be the Dirichlet form given by the closure of the quadratic form of the generator L_{2} of the semigroup in 𝕃_{2}. (For the Hilbert space theory of Dirichlet forms we refer to [4], [12], [16] and [20]; see also references therein.) More generally, for q ∈ (1, ∞) , we also define the generalized Dirichlet forms

E_{q}(g, g) = < ϕ_{p}(g), −L_{q}g > ·∥g∥_{𝕃_{q}}^{q−2},

with a dense domain containing D(L_{q}) ⊂ 𝕃_{q} and \frac{1}{p} + \frac{1}{q} = 1. Because P_{t} is contractive in 𝕃_{q} so, by the Lumer-Phillips theorem,

E_{q}(g, g) ≥ 0

for every elements g for which E_{q}(g, g) is finite.

Some Properties of Hypercontractive Semigroups

Given a contractive and positive semigroup P_{t}, t ≥ 0 on all 𝕃_{p} spaces, we would like now to discuss the hypercontractivity property. We remark that, (similarly to commutative case as in [22]), it is sufficient to consider hypercontractivity on positive cones because of the following result.

Proposition 3.2. If P_{t}, t ≥ 0 is positive and for any f ∈ 𝕃_{q(0)}^{+} we have

∥f_{t}∥_{q} ≤ C∥f∥_{q(0)}

with some 1 < q(0) < q < ∞ and C ≡ C(t, q(0), q) ∈ (0, ∞), then the same remains true for any f ∈ 𝕃_{q(0)}^{R}.

Proof : By Proposition 2.3 , we have that ±f ≤ |f|_{q(0)} and because P_{t} is a positive semigroup, we get

±P_{t}(f) ≤ P_{t}(|f|_{q(0)})  (100)

For f_{t} ≡ P_{t}f and its normalized tangent functional ϕ̃_{p}(f_{t}) ≡ ϕ_{p}(f_{t})/∥f_{t}∥_{q}, with complementary p and q, we define

f_{t}^{+} = \frac{|f_{t}|_{q} + f_{t}}{2} , f_{t}^{−} = \frac{|f_{t}|_{q} − f_{t}}{2}

and

ϕ̃_{p}^{+}(f_{t}) = \frac{|ϕ̃_{p}(f_{t})|_{p} + ϕ̃_{p}(f_{t})}{2}, ϕ̃_{p}^{−}(f_{t}) = \frac{|ϕ̃_{p}(f_{t})|_{p} − ϕ̃_{p}(f_{t})}{2}

By Proposition 2.3 , these functions are positive elements from 𝕃_{q}^{+} and 𝕃_{p}^{+}, respectively, for \frac{1}{p} + \frac{1}{q} = 1 . Because q-modulus is an isometry and as one can see ϕ̃_{p}(|f_{t}|_{q}) = |ϕ̃_{p}(f_{t})|_{p}, we have

< ϕ̃_{p}(f_{t}), f_{t} > = < |ϕ̃_{p}(f_{t})|_{p}, |f_{t}|_{q} >

what implies

< ϕ̃_{p}^{+}(f_{t}), f_{t}^{−} > = < ϕ̃_{p}^{−}(f_{t}), f_{t}^{+} >= 0

Hence, using (100), we get

∥f_{t}∥_{q} =< ϕ̃_{p}^{+}(f_{t}), f_{t} > + < ϕ̃_{p}^{−}(f_{t}), (−f_{t}) > ≤

< ϕ̃_{p}^{+}(f_{t}), P_{t}(|f|_{q(0)}) > + < ϕ̃_{p}^{−}(f_{t}), P_{t}(|f|_{q(0)}) > = < |ϕ̃_{p}(f_{t})|_{p}, P_{t}(|f|_{q(0)}) >

Because ∥|ϕ̃_{p}(f_{t})|_{p}∥_{p} = ∥ϕ̃_{p}(f_{t})∥_{p} = 1 so, by duality,

∥f_{t}∥_{q} ≤ ∥P_{t}(|f|_{q(0)})∥_{q} ≤ C(t, q)∥|f|_{q(0)}∥_{q(0)} = C(t, q)∥f∥_{q(0)} □

Next we observe that it is enough to consider only the case q(0) = 2 and q > 2.

Proposition 3.3. Suppose for some τ ∈ (0, ∞) we have ∥P_{τ}∥_{4, 2} ≤ C and ∥P_{τ}∥_{∞, ∞} ≤ 1. Then

∥P_{t}∥_{q(t), 2} ≤ exp \{2d(\frac{1}{2} − \frac{1}{q(t)})\}  (101)

with q(t) = 1 + e^{2t/c}, where c ≡ 4τ, and d ≡ log C.

The proof based on the Riesz-Thorin interpolation theorem and is similar to the one in the commutative case.

Next we recover the Simon’s result saying that strict hypercontractivity implies the existence of a spectral gap at the top of the spectrum of the generator.

Theorem 3.4. (Strict Hypercontractivity ⟹Spectral Gap) Let P_{t} be a strictly hypercontractive, 𝕃_{2}-symmetric and Markov-Feller semigroup in 𝕃_{p}^{R}(ω) spaces. Then

\frac{log 3}{2τ_{2,4}} < f− < 1, f >, f− < 1, f >>_{ω} ≤ E_{2}(f, f)

for all f ∈ A^{R} ∩ D(L).

Proof : For f = f^{∗} ∈ A let f̂ ≡ f− < 1, f > ≡ f − z and set f̃_{t} ≡ P_{t}f̂. We have

∥P_{t}f∥_{4}^{4} = z^{4} + 2z^{2} Tr [2ρ^{\frac{1}{2}} (ρ^{1/8}f̃_{t}ρ^{1/8})^{2} + ρ^{\frac{1}{4}} (ρ^{1/8}f̃_{t}ρ^{1/8})ρ^{\frac{1}{4}} (ρ^{1/8}f̃_{t}ρ^{1/8})]

+4z Tr [ρ^{\frac{1}{4}} (ρ^{1/8}f̃_{t}ρ^{1/8})^{3}] + ∥f̃_{t}∥_{4}^{4}  (102)

On the other hand

∥f∥_{2}^{4} = (∥f̃∥_{2}^{2} + z^{2})^{2} = z^{4} + 2z^{2}∥f̃∥_{2}^{2} + ∥f̃∥_{2}^{4}

Hence ∥P_{t}f∥_{4}^{4} ≤ ∥f∥_{2}^{4} implies the following inequality

2z^{2}Tr [2ρ^{\frac{1}{2}} (ρ^{1/8}f̃_{t}ρ^{1/8})^{2} + ρ^{\frac{1}{2}} f̃_{t}ρ^{\frac{1}{2}} f̃_{t}] + 4zTr [ρ^{\frac{1}{4}} (ρ^{1/8}f̃_{t}ρ^{1/8})^{3}] + ∥f̃_{t}∥_{4}^{4}

≤ 2z^{2}∥f̃∥_{2}^{2} + ∥f̃∥_{2}^{4}  (103)

Dividing both sides of (103) by 2z^{2} and taking the limit z → ∞ we get

∥f̃∥_{2}^{2} ≥ 2Tr (ρ^{3/4}f̃_{t}ρ^{\frac{1}{4}} f̃_{t}) + < f̃_{t}, f̃_{t} >

Since, (see Exercise below),

Tr(ρ^{3/4}F^{∗}ρ^{\frac{1}{4}} F) ≧ Tr(ρ^{\frac{1}{2}} F^{∗}ρ^{\frac{1}{2}} F)

we get ∥f̃∥_{2}^{2} ≥ 3∥f̃_{t}∥_{2}^{2} for every t ≥ τ_{2,4}. By the semigroup property this implies

∥P_{nτ_{2,4}} f̃∥_{2}^{2} ≤ 3^{−n}∥f̃∥_{2}^{2}

and so

− \frac{1}{n} log ∥f̃_{nτ_{2,4}}∥_{2}^{2} ≥ log 3 − \frac{log ∥f̃∥_{2}^{2}}{n}

Setting t = nτ_{2,4} and taking the limit t → ∞ we obtain that

lim_{t→∞}(−\frac{1}{t} log ∥f̃_{t}∥_{2}) ≥ \frac{log 3}{2τ_{2,4}}

Hence, by the spectral theorem,

\frac{log 3}{2τ_{2,4}} < f̃, f̃ >_{ω}≤ E_{2}(f, f)

for f ∈ A^{R} ∩ D(L). □

Exercise : Prove that for b(s) ≡ Tr ρ^{s}f̃_{t}ρ^{1−s}f̃_{t}, s ∈ [0, 1] one has b(3/4) ≥ b(1/2) . Hint : Note that b(s) is symmetric with respect to s = \frac{1}{2} and Reflection Positive, that is for every n ∈ ℕ , any z_{1}, ..., z_{n} ∈ ℂ and any s_{1}, ..., s_{n} ∈ [0, 1/2] the following inequality holds:

∑_{i,j=1}^{n} \overline{z_{i}}z_{j} b(s_{i} + s_{j}) ≥ 0

Using (102) one can show that the analog of Glimm’s lemma is also true, that is (101) together with the spectral gap implies strict hypercontractivity.

Theorem 3.5 :

(Hypercontractivity + Spectral Gap ⟹ Strict Hypercontractivity) Let P_{t} be a 𝕃_{2}-symmetric and Markov-Feller semigroup in 𝕃_{p}^{R}(ω) spaces such that with some constants C, T ∈ (0, ∞) one has for all t > T

∥P_{t}f∥_{4} ≦ C∥f∥_{2}  (104)

and with some m ∈ (0, ∞)

m < f− < 1, f >, f− < 1, f >>_{ω} ≤ E_{2}(f, f)  (105)

for all f ∈ A^{R} ∩ D(L). Then P_{t} is a strictly hypercontractive

Proof : For f = f^{∗} ∈ A, let f̂ ≡ f− < 1, f > and set f̃_{t} ≡ P_{t}f̂. Setting z ≡< 1, f > , we have

∥P_{t}f∥_{4}^{4} = z^{4} + 2z^{2}Tr [2ρ^{\frac{1}{2}} (ρ^{1/8}f̃_{t}ρ^{1/8})^{2} + ρ^{\frac{1}{4}} (ρ^{1/8}f̃_{t}ρ^{1/8})ρ^{\frac{1}{4}} (ρ^{1/8}f̃_{t}ρ^{1/8})] (106)

+4zTr [ρ^{\frac{1}{4}} (ρ^{1/8}f̃_{t}ρ^{1/8})^{3}] + ∥f̃_{t}∥_{4}^{4}

Using Hölder inequality for traces we have

Tr (ρ^{\frac{1}{2}} (ρ^{1/8}f̃_{t}ρ^{1/8})^{2}) ≦ ∥f̃_{t}∥_{4}^{2}
Tr (ρ^{\frac{1}{4}} (ρ^{1/8}f̃_{t}ρ^{1/8})ρ^{\frac{1}{4}} (ρ^{1/8}f̃_{t}ρ^{1/8})) ≦ ∥f̃_{t}∥_{4}^{2}
Tr [ρ^{\frac{1}{4}} (ρ^{1/8}f̃_{t}ρ^{1/8})^{3}] ≦ ∥f̃_{t}∥_{4}^{3}

Using (106) together with (107), we get

∥P_{t}f∥_{4}^{4} ≦ z^{4} + z^{2}6∥f̃_{t}∥_{4}^{2} + 4z∥f̃_{t}∥_{4}^{3} + ∥f̃_{t}∥_{4}^{4}  (107)

whence by our hypercontractivity assumption (104) for t > 2T

∥P_{t}f∥_{4}^{4} ≦ z^{4} + z^{2} 6C^{2}∥f̃_{t/2}∥_{2}^{2} + z 4C^{3}∥f̃_{t/2}∥_{2}^{3} + C^{4}∥f̃_{t/2}∥_{2}^{4} (108)

This together with the spectral gap assumption (105) yields

∥P_{t}f∥_{4}^{4} ≦ z^{4} + z^{2} 6C^{2}e^{−mt}∥f̃∥_{2}^{2} + z 4C^{3}e^{−m3t/2}∥f̃∥_{2}^{3} + C^{4}e^{−m2t}∥f̃∥_{2}^{4}
≦ z^{4} + z^{2} 8C^{2}e^{−mt}∥f̃∥_{2}^{2} + 3C^{4}e^{−m2t}∥f̃∥_{2}^{4}

Hence choosing T_{1} ∈ (0, ∞) such that 8C^{2}e^{−mT_{1}} ≦ 2 and 3C^{4}e^{−m2T_{1}} ≦ 1 , we arrive at

∥P_{t}f∥_{4}^{4} ≦ (z^{2} + ||f̃∥_{2}^{2} )^{2} = ∥f∥_{2}^{4}

for all t > 2 max (T, T_{1}) , from which the strict hypercontractivity follows. □

4. Entropy and Infinitesimal Description of Hypercontractivity.

To study the differentiability properties of 𝕃_{r} norms with respect to r, we introduce a family of maps I_{p,q} , p, q ∈ (1, ∞) defined on A^{+} as follows

I_{p,q}(f) = ρ^{− \frac{1}{2p}} (ρ^{\frac{1}{2q}} fρ^{\frac{1}{2q}})^{q/p}ρ^{− \frac{1}{2p}}  (109)

One has the following result :

Theorem 4.1. For p, q ∈ (1, ∞) , I_{p,q} extends to a continuous bijective map from 𝕃_{q}^{+}(ω) onto 𝕃_{p}^{+}(ω) with the following properties

(i) ∥I_{p,q}(f)∥_{𝕃_{p}(ω)}^{p} = ∥f∥_{𝕃_{q}(ω)}^{q} (ii) I_{p,p} = id, I_{p,r} = I_{p,q}I_{q,r} ∀p, q, r ∈ [1, ∞) (iii) I_{p,q}(cf) = c^{q/p}I_{p,q}(f), ∀c ≥ 0 (iv) If \frac{1}{p} + \frac{1}{q} = 1 and ∥f∥_{𝕃_{q}(ω)} = 1, then I_{p,q}(f) = ϕ_{p}(f).

Proof : The properties (i) - (iv) can be directly shown on A^{+} using the formula (109), and are left as an Exercise. To show that I_{p,q} can be extended to bijective and continuous maps from 𝕃_{q}^{+}(ω) onto 𝕃_{p}^{+}(ω), we consider first the case q < p . If a_{n} ∈ A^{+} and a_{n} → f in 𝕃_{q}(ω) , then

∥I_{p,q}(a_{n}) − I_{p,q}(a_{m})∥ _{𝕃_{p}(ω)} = ∥(ρ^{\frac{1}{2q}} a_{n}ρ^{\frac{1}{2q}})^{q/p} − (ρ^{\frac{1}{2q}} a_{m}ρ^{\frac{1}{2q}})^{q/p}∥_{𝕃_{p}(Tr)}

≤ ∥ |ρ^{\frac{1}{2q}} a_{n}ρ^{\frac{1}{2q}} − ρ^{\frac{1}{2q}} a_{m}ρ^{\frac{1}{2q}} |^{q/p}∥ _{𝕃_{p}(Tr)} = ∥a_{n} − a_{m}∥_{𝕃_{q}(ω)}^{q/p} →_{n,m→∞} 0

In the above we used the estimation given in [5]. It follows that I_{p,q}(a_{n}) is convergent in 𝕃_{p}(ω). Thus we can define I_{p,q}(f) = lim_{n→∞} I_{p,q}(a_{n}) and get a continuous map I_{p,q} : 𝕃_{q}^{+}(ω) →𝕃_{p}^{+}(ω). For q > p we proceed as follows. It is clear that for a_{n} ∈ A^{+}, a_{n} → 0 in 𝕃_{q}(ω), we have ∥I_{p,q}(a_{n})∥_{𝕃_{p}(ω)} → 0. So it is enough to consider only convergent sequences {a_{n}} such that ∥a_{n}∥_{𝕃_{q}(ω)} = 1. Let q′ and p′ be complementary to q and p, respectively. If q′ < p, then

lim_{n→∞} I_{p,q}(a_{n}) = lim_{n→∞} I_{p,q′}I_{q′,q}(a_{n}) = lim_{n→∞} I_{p,q′}(ϕ_{q′}(a_{n}))

= I_{p,q′}(ϕ_{q′}(f))

and so I_{p,q}(f) is well defined. If q′ > p, then q < p′ and we have

lim_{n→∞} I_{p,q}(a_{n}) = lim_{n→∞} I_{p,p′}I_{p′,q}(a_{n}) = ϕ_{p}(I_{p′,q}(f))

Thus I_{p,q} : 𝕃_{q}^{+}(ω) → 𝕃_{p}^{+}(ω) is continuous for arbitrary p and q. Because

I_{p,q}I_{q,p}|_{A^{+}} = id|_{A^{+}}

so I_{p,q}I_{q,p} = id on 𝕃_{p}^{+}(ω) and I_{p,r} = I_{p,q}I_{q,r} on 𝕃_{r}^{+}(ω). This completes the proof of the theorem. □

Entropy and Differentiability of the 𝕃_{p}(ω) norms

Using the maps I_{p,q} we define the curves which flow through the cones 𝕃_{p}^{+}(ω), as follows:

z_{q,f}(u) = I_{q+u,q}(f), f ∈ 𝕃_{q}^{+}(ω), u ≥ 0

Because 𝕃_{q+u}^{+}(ω) ⊂ 𝕃_{q}^{+}(ω), we can consider u → z_{q,f}(u) as a curve in one Banach space 𝕃_{q}(ω). Let f ∈ A^{+} be invertible. Then

\frac{d}{du}z_{q,f}(u)|_{u=0} = \frac{d}{du}(ρ^{− \frac{1}{2(q+u)}} (ρ^{\frac{1}{2q}} fρ^{\frac{1}{2q}})^{q/(q+u)}ρ^{− \frac{1}{2(q+u)}} )|_{u=0}

exists. We define

T_{q}(f) = −q \frac{d}{du}z_{q,f}(u)|_{u=0}  (110)

whenever the corresponding derivative exists. The function T_{q} : D(T_{q}) → 𝕃_{q}^{R}(ω) is defined on a dense domain D(T_{q}) ⊃ A^{+} for all q > 1. By direct calculation for f ∈ A^{+} one gets the following formula called the renormalized Operator Valued (Relative) Entropy

T_{q}(f) = ρ^{− \frac{1}{2q}} ((ρ^{\frac{1}{2q}} fρ^{\frac{1}{2q}}) log(ρ^{\frac{1}{2q}} fρ^{\frac{1}{2q}})) ρ^{− \frac{1}{2q}} − \frac{1}{2q} \{f, log ρ\} (111)

where we have set {g, h} ≡ gh + hg. In commutative algebra when each f commutes with ρ the OVE is independent of ρ and simply equals f log f.

For further application we observe that one has the following covariance relation between T_{2} and T_{q} (which generalises the corresponding property of classical relative entropy).

Theorem 4.2 (Covariance Property) If \frac{1}{p} + \frac{1}{q} = 1 and f ∈ A^{+}, then

< I_{p,q}(f), T_{q}(f) >_{ω} = \frac{2}{q} < I_{2,q}(f), T_{2}(I_{2,q}(f)) >_{ω}  (112)

Proof : We have

\frac{2}{q} < I_{2,q}(f), T_{2}(I_{2,q}(f)) >_{ω}=

= \frac{2}{q}Tr (ρ^{\frac{1}{2}} I_{2,q}(f)ρ^{\frac{1}{2}} (I_{2,q}(f)ρ^{\frac{1}{4}} (log ρ^{\frac{1}{4}} I_{2,q}(f)ρ^{\frac{1}{4}} ) ρ^{− \frac{1}{4}} − \frac{1}{4} \{I_{2,q}(f) , log ρ\}))

= \frac{2}{q}Tr (ρ^{\frac{1}{4}} (ρ^{\frac{1}{2q}} fρ^{\frac{1}{2q}})^{q} (log(ρ^{\frac{1}{2q}} fρ^{\frac{1}{2q}})^{\frac{q}{2}} ) ρ^{− \frac{1}{4}} )

− \frac{1}{2q}Tr (ρ^{\frac{1}{4}} (ρ^{\frac{1}{2q}} fρ^{\frac{1}{2q}})^{\frac{q}{2}} (ρ^{\frac{1}{2q}} fρ^{\frac{1}{2q}})^{\frac{q}{2}} ρ^{− \frac{1}{4}} log ρ)

− \frac{1}{2q}Tr (ρ^{\frac{1}{4}} (ρ^{\frac{1}{2q}} fρ^{\frac{1}{2q}})^{\frac{q}{2}} ρ^{\frac{1}{4}} (log ρ)ρ^{− \frac{1}{4}} (ρ^{\frac{1}{2q}} fρ^{\frac{1}{2q}})^{\frac{q}{2}} ρ^{− \frac{1}{4}} )

= Tr ((ρ^{\frac{1}{2q}} fρ^{\frac{1}{2q}})^{q} (log ρ^{\frac{1}{2q}} fρ^{\frac{1}{2q}} − \frac{1}{q} log ρ)) (113)

On the other hand, if \frac{1}{p} + \frac{1}{q} = 1, using formula (111), we have

< I_{p,q}(f), T_{q}(f) >_{ω}=Tr(ρ^{\frac{1}{2}} ρ^{− \frac{1}{2p}} (ρ^{\frac{1}{2q}} fρ^{\frac{1}{2q}})^{q−1}ρ^{− \frac{1}{2p}} ρ^{\frac{1}{2}} (fρ^{\frac{1}{2q}} (log ρ^{\frac{1}{2q}} fρ^{\frac{1}{2q}})ρ^{− \frac{1}{2q}} (114)

− \frac{1}{2q} \{f , log ρ\}))

which can be shown to be equal to the formula on the right hand side of (113) by simple manipulations under the trace. □ The above introduced map T_{q} is useful to measure the speed of the change of 𝕃_{q}(ω) norms with respect to q.

Theorem 4.3 (Differentiation of 𝕃_{p} Norm). If f ∈ A^{+}, then q → ∥f∥ _{𝕃_{q}} is differentiable and

\frac{d}{dq} ∥f∥_{𝕃_{q}}^{q} =< I_{p, q}(f), T_{q}(f) >_{ω}  (115)

with p being complementary to q. Thus for any continuously differentiable map q ≡ q(t) with values in (1, ∞), we have

\frac{d}{dt}∥f∥_{𝕃_{q}}^{q} = q̇ < I_{p, q}(f), T_{q}(f) >_{ω}  (116)

with q̇ ≡ \frac{d}{dt}q.

Proof : Let f ∈ A^{+} be invertible. Setting A_{q} ≡ ρ^{\frac{1}{2q}} fρ^{\frac{1}{2q}}, we have

∥f∥_{𝕃_{q}}^{q} = Tr(A_{q}^{q})  (117)

To show the desired differentiability, first we need to find a convenient representation of the difference of quantities (117) corresponding values at q + △q an q. Because A_{q} is positive and invertible, we have

Tr A_{q+△q}^{q+△q} − Tr A_{q}^{q} =Tr(∫_{0}^{1} ds\frac{d}{ds}A_{q+△q}^{s(q + △q)}A_{q}^{(1−s)q})

=Tr(∫_{0}^{1} ds\frac{d}{ds}e^{s(q + △q) log A_{q+△q}}e^{(1−s)q log A_{q}})

Since the corresponding derivative is given by

A_{q+△q}^{s(q + △q)} ((q + △q) log A_{q+△q} − q log A_{q}) A_{q}^{(1−s)q}

after simple transformations we arrive at

Tr A_{q+△q}^{q+△q} −Tr A_{q}^{q} = (q + △q)Tr(∫_{0}^{1} dsA_{q}^{(1−s)q}A_{q+△q}^{s(q + △q)}[log A_{q+△q} − log A_{q}])

+ △q Tr(∫_{0}^{1} dsA_{q}^{(1−s)q}A_{q+△q}^{s(q + △q)} log A_{q})

Dividing it by △q and taking the limit △q → 0, we obtain

\frac{d}{dq}TrA_{q}^{q} = q Tr (A_{q}^{q}(\frac{d}{dq} log A_{q})) +TrA_{q}^{q} log A_{q}  (118)

To compute the derivative of the logarithmic term we use the following integral representation

log A_{q} = ∫_{0}^{∞}dλ (\frac{1}{λ} − \frac{1}{λ + A_{q}} )

From this we obtain

\frac{d}{dq} log A_{q} = ∫_{0}^{∞}dλ (\frac{1}{λ + A_{q}} ) \frac{dA_{q}}{dq} (\frac{1}{λ + A_{q}} )

and hence the formula (118) can be represented as follows

\frac{d}{dq}TrA_{q}^{q} = q TrA_{q}^{q−1}(\frac{d}{dq} A_{q}) + TrA_{q}^{q} log A_{q}

Since

\frac{dA_{q}}{dq} = −\frac{1}{2q^{2}} \{A_{q} , log ρ\}

we arrive at

\frac{d}{dq} ∥f∥_{𝕃_{q}}^{q} ≡ \frac{d}{dq}TrA_{q}^{q} = −\frac{1}{q}TrA_{q}^{q} log ρ + TrA_{q}^{q} log A_{q}

= (Tr(ρ^{\frac{1}{2q}} fρ^{\frac{1}{2q}})^{q} (log ρ^{\frac{1}{2q}} fρ^{\frac{1}{2q}} − \frac{1}{q} log ρ))

By comparing this with (114) and the right hand side of (113) we see that the formula in the bracket equals < I_{p,q}(f), T_{q}(f) >_{ω}. This completes the proof for invertible f ∈ A^{+}. Since such f are dense in the operator norm in A^{+}, to finish the proof of the first part it suffices to notice that both sides of equation (116) are continuous with respect to f in the operator norm. From that the second part of the theorem easily follows. □ By similar arguments one can show the following.

Corollary 4.4. For any f ∈ A^{R}, we have

\frac{d}{dt}∥f∥_{𝕃_{q(t)}}^{q(t)} = q̇(t) < I_{p(t),q(t)}(|f|_{q(t)}), T_{q(t)}(|f|_{q(t)}) >_{ω}  (119)

Hypercontractivity and Logarithmic Sobolev Inequality

Similarly as for the contractivity condition, one would like to introduce the infinitesimal condition for hypercontractivity. To this end it useful to introduce the following definition.

Definition 4.5. We say that the LS_{q} - Logarithmic Sobolev inequality is true iff for c(q) ≡ \frac{q}{2(q−1)}c and d(q) ≡ \frac{2}{q}d defined with some constant c ∈ (0, ∞) and d ∈ [0, ∞), respectively, we have

< I_{p,q}(f), T_{q}(f) > −∥f∥_{q}^{q} log ∥f∥_{q} ≤ c(q)E_{q}(f, f) + d(q)∥f∥_{q}^{q} (120)

for all f ∈ 𝕃_{q}^{+} for which the right hand side is finite, where \frac{1}{p} + \frac{1}{q} = 1.

We recall that by Theorem 4.2 we have the following covariance property of relative entropy

< I_{p,q}(f), T_{q}(f) > = \frac{2}{q} < I_{2,q}(f), T_{2}(I_{2,q}(f)) >  (121)

provided \frac{1}{p} + \frac{1}{q} = 1. Hence we see that solely LS_{2} implies

< I_{p,q}(f), T_{q}(f) > − ∥f∥_{q}^{q} log ∥f∥_{q} ≤ \frac{2c}{q}E_{2}(I_{2,q}(f), I_{2,q}(f)) + \frac{2d}{q}∥f∥_{q}^{q} (122)

Later on the inequality LS_{2} with given coefficients c and d will be briefly called the inequality LS(c,d). It is well known that in commutative case LS(c,d) actually implies LS_{q}. For that one needs some extra regularity of Dirichlet forms which we introduce in the following definition. Recall that for q ∈ (1, ∞) one defines the generalized Dirichlet forms by

E_{q}(g, g) =< ϕ_{p}(g), −L_{q}g > ·∥g∥_{𝕃_{q}}^{q−2},

with \frac{1}{p} + \frac{1}{q} = 1 defined on a dense domain containing the domain D(L_{q}) ⊂ 𝕃_{q} of the generator L_{q} of the Markov semigroup extended to 𝕃_{q} . Note that if g is positive, then

E_{q}(g, g) = < I_{p,q}(g), −L_{q}g >

Definition 4.6. We say that the Dirichlet forms E_{p} are 𝕃_{p} - regular iff there is d_{0} ≥ 0 such that ∀ q ≥ 2 and g ∈ D(E_{q}) we have

E_{2}(I_{2,q}(g), I_{2,q}(g)) ≤ \frac{q^{2}}{4(q − 1)}E_{q}(g, g) + d_{0}∥g∥_{q}^{q}  (123)

Remark : Note that the classical Dirichlet forms in commutative L_{p} spaces satisfy this regularity property with d_{0} = 0. Some noncommutative examples will be given at the end of this lecture. The following lemma plays a crucial role for the infinitesimal description of hypercontractivity.

Lemma 4.7. Let f ∈ A^{+} ∩ D(E_{q_{0}}) be strictly positive. Then, with q ≡ q(t) ≡ 1 + (q_{0} − 1)e^{2t/c}, we have

\frac{d}{dt} log ∥f_{t}∥_{q(t)}_{|_{t=0}} =  (124)

= \frac{q̇_{0}/q_{0}}{∥f∥_{q}^{q}} \{(⟨I_{p,q}(f), T_{q}(f)⟩ − ∥f∥_{q}^{q} log ∥f∥_{q}) − \frac{qc}{2(q − 1)}E_{q}(f, f)\}

with p ≡ p(t) being complementary to q ≡ q(t) ∈ (1, ∞).

Lengthy proof of this lemma is omitted here , (but can be found in [36]). Assuming this lemma we would like to prove the following result which provides an infinitesimal characterisation of hypercontractivity property in the noncommutative 𝕃_{p} spaces.

Theorem 4.8. Suppose P_{t} is a 𝕃_{2}-symmetric Feller semigroup which is hypercontractive, that is we have

∥P_{t}∥_{q(t), 2} ≤ exp \{2d(\frac{1}{2} − \frac{1}{q(t)})\}  (125)

with d ∈ [0, ∞) and q(t) = 1 + e^{2t/c} defined with some constant c ∈ (0, ∞). Then the following LS(c,d) Logarithmic Sobolev inequality is true.

< f, T_{2}(f) > − ∥f∥_{𝕃_{2}}^{2} log ∥f∥_{𝕃_{2}} ≤ c E_{2}(f, f) + d ∥f∥_{𝕃_{2}}^{2} (126)

with some constants c ∈ (0, ∞) and d ∈ [0, ∞) for all f ∈ 𝕃_{2}^{+} for which the right hand side is finite. Conversely, if LS(c,d) is true and additionally the forms E_{p} are 𝕃_{p} - regular with some constant d_{0} ∈ (0, ∞), then (125) holds with the constant d replaced by d + c · d_{0}.

Proof : Assuming Lemma 4.7 we prove first the implication of LS(c,d) from hypercontractivity. To this end we consider a function

F (t) = exp \{−2d(\frac{1}{2} − \frac{1}{q(t)})\} · ∥f_{t}∥_{q(t)}

for f ∈ A^{+} ∩ D(L). It follows from our assumption (125) that we have log F (t) ≤ log F (0). Therefore the corresponding right derivative at t = 0 has to satisfy

\frac{d log F (t)}{dt}|_{t=0} = −2d\frac{q̇(0)}{q(0)^{2}} + \frac{d}{dt} log ∥f_{t}∥_{q(t)}|_{t=0} ≤ 0  (127)

Using Lemma 4.7 with q_{0} = 2 for t = 0, we have p_{0} = 2 and I_{p_{0},q_{0}} = id and therefore

\frac{d}{dt} log ∥f_{t}∥_{q(t)}|_{t=0} = \frac{1}{c} \frac{1}{∥f_{t}∥_{2}^{2}} (< f_{t}, T_{2}(f_{t}) > − ∥f_{t}∥_{2}^{2} log ∥f_{t}∥_{2} − cE_{2}(f_{t}, f_{t}))

This together with (127) yields LS(c,d).

To prove the second part we use Lemma 4.7 together with LS(c,d) inequality, to get

\frac{d}{dt} log ∥f_{t}∥_{q(t)}

= \frac{q̇/q}{∥f_{t}∥_{q}^{q}} \{(⟨I_{p,q}(f_{t}), T_{q}(f_{t})⟩ − ∥f_{t}∥_{q}^{q} log ∥f_{t}∥_{q}) − \frac{q c}{2(q − 1)}E_{q}(f_{t}, f_{t})\}

≤ \frac{q̇/q}{∥f_{t}∥_{q}^{q}} \{(\frac{2c}{q}E_{2}(I_{2,q}(f_{t}), I_{2,q}(f_{t})) + \frac{2}{q} d∥I_{2,q}(f_{t})∥_{2}^{2} ) − \frac{q c}{2(q − 1)}E_{q}(f_{t}, f_{t})\} (128)

Since from our assumption of 𝕃_{p} -regularity we have

\frac{2}{q}E_{2}(I_{2,q}(f_{t}), I_{2,q}(f_{t})) ≤ \frac{q}{2(q − 1)} E_{q}(f_{t}, f_{t}) + \frac{2}{q} d_{0}∥f_{t}∥_{q}^{q}

using (128) , we obtain

\frac{d}{dt} log ∥f_{t}∥_{q(t)} ≤ \frac{2q̇}{q^{2}} (d + c · d_{0})

Integration of this differential inequality with q(0) = 2 leads to

∥f_{t}∥_{q(t)} ≤ ∥f∥_{q(0)} exp \{2(d + c · d_{0})(\frac{1}{2} − \frac{1}{q(t)})\}

for any f in a dense set in 𝕃_{2}^{+}. From this the desired result easily follows by continuity. □

Further Properties of the Logarithmic Sobolev Inequality

In this Section we study further properties of Logarithmic Sobolev inequality and Hypercontractivity in noncommutative 𝕃_{p} spaces. In particular we prove the Rothaus implication of the Spectral Gap inequality from LS, (corresponding to Simon’s complement - Theorem 3.4 above). We also show the Rothaus lemma on improvement of Logarithmic Sobolev inequality as a consequence of LS(c,d), Lemma 2.9 and the Spectral Gap; (this result corresponds to the Glimm lemma on strict hypercontractivity).

Now we would like to show that LS(c, d=0) inequality implies spectral gap property for the generator L_{2}.

Theorem 4.10 Let P_{t} be a 𝕃_{2}-symmetric Markov-Feller semigroup in 𝕃_{p}(ω) spaces. Suppose that for every f ∈ A^{+} ∩ D(L) the LS(c, d=0) inequality holds. Then

\frac{1}{2c} < f − < 1, f >, f − < 1, f >> ≤ E_{2}(f, f)

The idea of the proof of Theorem 4.10 is similar to the one in commutative case. Analogously as in commutative case one also shows that the spectral gap property can be used to improve the LS inequality.

Theorem 4.11. Let ω(f) = Tr(ρf). Suppose LS(c,d) inequality holds for every f ∈ A^{+} ∩D(L) and moreover that the following spectral gap inequality is true

m∥f − < 1, f > ∥_{2}^{2} ≤ E_{2}(f, f)

with some m ∈ (0, ∞). Then LS(c’, d’=0) inequality holds with c′ ≡ c + (d + 1)/m.

The remaining key condition of the Logarithmic Sobolev inequality - the product property- will be considered in the next lecture.

The 𝕃_{p} Regularity of Dirichlet Forms

In this Section we describe the results concerning the 𝕃_{p} regularity of Dirichlet forms which plays the crucial role in the proof of the integration lemma (Theorem 4.8). In particular we note that the generalised Dirichlet form of any bounded generator and the generator of any subordinated semigroup possess the 𝕃_{p} regularity. Additionally this property is also true for if the underlying state is a trace and the generator is symmetric in the corresponding 𝕃_{2} space. Details of the proofs are provided in the Appendix II.

Recall that by Definition 4.6 the E_{p} form is called 𝕃_{p} regular iff there is d_{0} ≥ 0 such that ∀ q ≥ 2 and g ∈ D(E_{q}) we have

E_{2}(I_{2,q}(g), I_{2,q}(g)) ≤ \frac{q^{2}}{4(q − 1)} E_{q}(g, g) + d_{0}∥g∥_{q}^{q}

where E_{q}(g, g) ≡ < ϕ_{p}(g), −L_{q}g > ·∥g∥_{𝕃_{q}}^{q−2} = < I_{p,q}(g), −L_{q}g > with \frac{1}{p} + \frac{1}{q} = 1. Note that replacing g by I_{q,2}(f) and using Theorem 4.1 we can rewrite the regularity condition as follows

E_{2}(f, f) ≤ \frac{q^{2}}{4(q − 1)} E_{2}(I_{p,2}(f), I_{q,2}(f)) + d_{0}∥f∥_{2}^{2}

Clearly that is equivalent to the following inequality:

lim_{t→0} \frac{1}{t} < f, (1 − P_{t})f > ≤ \frac{q^{2}}{4(q − 1)} lim_{t→0} \frac{1}{t} < I_{p,2} (f), (1 − P_{t})I_{q,2} (f) >
+ d_{0}∥f∥_{2}^{2}

where f ∈ 𝕃_{2}^{+} ∩ D(L_{2}) is such that I_{q,2}(f) ∈ D(L_{q}) and \frac{1}{p} + \frac{1}{q} = 1. The last is certainly satisfied if for all small t ≥ 0

< f, (1 − P_{t})f >≤ \frac{q^{2}}{4(q − 1)}(< I_{p,2} (f), I_{q,2} (f) > − < I_{p,2} (f), P_{t}I_{q,2} (f) >)

+td_{0}∥f∥_{2}^{2}

But < I_{p,2} (f), I_{q,2} (f) > = < I_{p,q}(I_{q,2} (f)), I_{q,2} (f) > = ∥I_{q,2} (f)∥_{q}^{q} = ∥f∥_{2}^{2} so we can rewrite the above inequality in the following form: < I_{p,2} (f), P_{t}I_{q,2} (f) > ≤ ξ_{q} < f, P_{t}f > + (1 − ξ_{q})∥f∥_{2}^{2} + ξ_{q}td_{0}∥f∥_{2}^{2} where ξ_{q} = 4(q − 1)/q^{2}. Thus we note that the regularity of interest to us can be represented as a property of the following positive function

h(q) = < I_{p,2} (f), P_{t}I_{q,2} (f) >

for q ∈ (1, ∞) . Detailed study of this function, (based on a notion of reflection positivity) gives the following three general results, (see Appendix II for detailed proofs). Theorem 4.12. (Bounded Generators have Regular Forms) If the generator L_{2} is bounded, then (150) holds with d_{0} = (∥L_{2}∥ + 1). Theorem 4.13. (Regularity in 𝕃_{p}(Tr) spaces) Suppose P_{t} is a 𝕃_{2}-symmetric and Markov-Feller semigroup in 𝕃_{p}(Tr) spaces. Then the corresponding generalized Dirichlet forms are 𝕃_{p} - regular. Generators of Subordinated Semigroups have 𝕃_{p} Regular Forms Let P_{t} be a subordinated semigroup to the modular automorphism

α_{τ}(f) ≡ ρ^{−iτ}fρ^{iτ}

that is

P_{t}(f) ≡ ∫ ν_{t}(dτ)α_{τ}(f)

with a measure ν_{t}(dτ) defined as the Fourier transform of exp{−tF (x)} with F given by the Lévy-Khinchine formula

F (x) = ax^{2} + ∫ (1 − cos xy) \frac{1 + y^{2}}{y^{2}} dμ(y)

where dμ is a positive and finite measure on R such that μ({0}) = 0. One can see that, for any z ∈ [0, 1], the semigroup P_{t} satisfies the following symmetry condition

< P_{t}f , g >_{ω,z} ≡ Tr (ρ^{1−z}P_{t}f^{∗}ρ^{z}g) = < f , P_{t}g >_{ω,z}

In the present situation we also have for any p, q ∈ [1, ∞)

α_{τ}I_{p,q}(f) = I_{p,q}(α_{τ}f)

which allows us to get the following result

Theorem 4.15. The generalized Dirichlet forms associated to a subordinated Feller - Markov semigroup are 𝕃_{p} regular.

Lecture IV

Particle Structure and Hypercontractivity

Introduction

In this lecture we would like to demonstrate that particle structure and hypercontractivity are intimately related. As a consequence this allows us to prove a product property of Logarithmic Sobolev inequality. Suppose P_{t} = e^{tL} is a symmetric Markov semigroup in 𝕃_{2}(μ). If its generator L has a discrete spectrum, the following representation holds

P_{t}f = ∑_{n} e^{−tλ_{n}}⟨ψ_{n}, f⟩ ψ_{n},

where ⟨·, ·⟩ denotes the scalar product in 𝕃_{2}(μ) and ψ_{n} is a normalised eigenfunction corresponding to an eigenvalue λ_{n}. Recall that the semigroup P_{t} is strictly hypercontractive iff there exists T ∈ (0, ∞) such that

∀t > T, ||P_{t}f||_{𝕃_{4}}^{4} ≤ ||f||_{𝕃_{2}}^{4}.

In terms of our spectral representation, we can write this inequality as follows

∑_{n_{1},..,n_{4}} e^{−t(λ_{n_{1}}+..+λ_{n_{4}})}μ(ψ_{n_{1}}..ψ_{n_{4}}) ∏_{l=1,..,4}⟨ψ_{n_{l}}, f⟩ ≤ ∑_{n_{1},n_{2}}⟨ψ_{n_{1}}, f⟩^{2} ⟨ψ_{n_{2}}, f⟩^{2}.

From this one can see that hypercontractivity is related to special properties of the spectrum and overlapping properties of the eigenstates. Namely, to have hypercontractivity in the considered case when the spectrum is discrete, the eigenvalues λ_{n} have to grow in a certain way to infinity and the random variables ψ_{n} have to be in some sense weakly dependent behaving similarly to the Gaussian random variables.

In this lecture we explore further the idea presented above in the noncommutative setting. In particular we explain a strategy generalising slightly the one of [38] (related to the Fock structure of the Gaussian case of the free field theory).

The key reference related to this lecture is [11], (see also [24], [47]).

The General Strategy

We begin from setting an abstract scene for hypercontractivity including an interpolating family {𝔹_{q}, || · ||_{q}}_{q∈[1,∞]} of Banach spaces such that

• 𝔹_{2} is a Hilbert space denoted by (H , ⟨· , ·⟩_{H}).

• For any p, q ∈ [1, ∞], p ≤ q, 𝔹_{q} is dense subset of 𝔹_{p} and for any f ∈ 𝔹_{q}

||f||_{p} ≤ ||f||_{q}.

• For any p, q ∈ (1, ∞), p < q, \frac{1}{p} + \frac{1}{q} = 1, we have 𝔹_{p} ⊃ H ⊃ 𝔹_{q} and the spaces 𝔹_{p} and 𝔹_{q} are dual to each other.

It will be convenient to consider also a Banach space {A, || · ||} such that A is a dense subset of 𝔹_{∞} . The primary examples we have in mind include the classical 𝕃_{p}(μ) and noncommutative 𝕃_{p}(ω) spaces , associated to a probability measure μ and a state ω on a non-commutative algebra, respectively, discussed in the previous lectures (cf [27]-[29], [36]). Let (P_{t} ≡ e^{tL})_{t≥0} be a strongly continuous semigroup in 𝔹_{∞} which is symmetric in H. Then, by general arguments it can be extended to all 𝔹_{q} spaces. A semigroup (P_{t})_{t≥0} is called contractive in the 𝔹_{q}-spaces iff for any q in [1, ∞]

||P_{t}f||_{q} ≤ ||f||_{q}.

A semigroup (P_{t})_{t≥0} is called hypercontractive iff there is T in (0, ∞) such that

∀t ≥ T, ||P_{t}f||_{4} ≤ ||f||_{2}.

Then by general arguments there is a constant c ∈ (0, ∞) such that

||P_{t}f||_{q} ≤ ||f||_{2}  (129)

for q = q(t) ≡ 1 + exp (\frac{2}{c}t).

In this framework the general strategy, (extending the arguments described in the introduction), can be formulated as follows

Theorem IV.1 :

Suppose that the following conditions are satisfied :

(i) 𝔹_{2} has the following decomposition into orthogonal subspaces

𝔹_{2} = ⊕_{N=0}^{∞}H_{N},

where the space H_{0} consists of constant functions and for any N ∈ ℕ and t ≥ 0

P_{t}(H_{N} ) ⊆ H_{N}

(ii) The spectrum of the generator L restricted to H_{N} is contained in (−∞, −ηN], for some η ∈ (0, ∞) independent of N.

(iii) For any f_{N} ∈ H_{N} the following Gaussian bound is true

∥f_{N}∥_{4} ≤ C^{N} ∥f_{N}∥_{2},

with C ∈ (0, ∞) independent of N and f_{N}.

Then the semigroup P_{t} is hypercontractive.

A similar result in the context of the Fock structure related the 𝕃_{p}(μ) spaces defined with a Gaussian measure μ and corresponding Dirichlet operator was proven in [38] (cf Theorem I.17). Formulating our result in a more general setting enable us to apply it to a broader class of nontrivial examples including some noncommutative spin systems (described below) and the one-dimensional Ising model with nearest neighbours interaction (cf [11] for details).

Proof : Suppose f ∈ 𝔹_{2}. Then, by assumption (i), f has the following decomposition

f = ∑_{N=0}^{∞} f_{N}.

Using this and the triangle inequality, we get

∥P_{t}f∥_{4} ≤ ∑_{N=0}^{∞} ∥P_{t}f_{N}∥_{4}.

for any t ≥ 0 . By (i) P_{t}f_{N} belongs also to H_{N}, so that using (iii) one gets

∥P_{t}f∥_{4} ≤ ∑_{N=0}^{∞} C^{N}∥P_{t}f_{N}∥_{2}.

This combined with hypothesis (ii) on the spectrum, implies

∥P_{t}f∥_{4} ≤ ∑_{N=0}^{∞} exp(−ηNt)C^{N}∥f_{N}∥_{2}.

The subspaces H_{N} are orthogonal, therefore ∥f_{N}∥_{2} ≤ ∥f∥_{2}. Combining all the above, we conclude with the following inequality

∥P_{t}f∥_{4} ≤ [∑_{N=0}^{∞} exp(−ηNt)C^{N}] ∥f∥_{2}.

If t ≥ T, for some sufficiently large T ∈ (0, ∞), the series on the right hand side is convergent. Then P_{t} has a bounded norm from 𝔹_{2} to 𝔹_{4}. The hypotheses of the theorem imply the existence of a spectral gap, which enables us to conclude that the semigroup is hypercontractive, (see Theorem 3.5 in Lecture III ) . ⋄

Gaussian Bounds in Product Spaces and Hypercontractivity

Suppose H_{i} are finite dimensional Hilbert spaces, indexed by elements i of a countable set S, which are isomorphic to a given finite dimensional Hilbert space H_{0}. Let 𝔹_{2} ≡ H ≡ ⊗_{i∈S}H_{i}. Given an orthonormal base {ϕ^{l}}_{l=0,..,n−1} in H_{0}, we introduce the isomorphic orthonormal base ϕ_{i}^{l(i)} in H_{i}. For a set X ⊂ S of finite cardinality |X| and a function K ≡ K(X), K : X → {1, .., n − 1}^{X}, we define a vector Φ_{X}^{K} ∈ H formally given by ⊗_{i∈X}ϕ_{i}^{K(i)} ⊗_{j∈S\X} ϕ_{j}^{0}. We set Φ_{0} ≡ ⊗_{j∈S}ϕ_{j}^{0} to be a vector corresponding formally to X = ∅ (with the convention K(∅)_{i} ≡ 0 for each i ∈ S). The family Φ_{X}^{K} , with X ⊂ S, |X| < ∞ and K ≡ K(X), forms an orthonormal base with respect to the scalar product < , >_{H} in H. Given N ∈ ℕ, we define the N-particle subspace H_{N} of H as the closure of Span{Φ_{X}^{K} : |X| = N}. Let Q : D_{4} → ℂ be a quartic form with a dense domain such that Span{Φ_{X}^{K}}_{X,K} ⊂ D_{4} ⊂ H^{⊗4} . We assume that Q satisfies the following natural conditions (met for noncommutative 𝕃_{p} spaces , see the Exercise before Theorem 2.1 in Section 2 of Lecture III). First of all Q is linear and anti-linear with respect to the first and third, and second and fourth argument, respectively, and such that

∥Ψ∥_{4}^{4} ≡ Q(Ψ, Ψ, Ψ, Ψ) ≥ 0

with equality only for the zero element. Moreover we assume that

(Qi)

Q(Ψ, Φ_{0}, Φ_{0}, Φ_{0}) = Q(Φ_{0}, Ψ, Φ_{0}, Φ_{0}) = Q(Φ_{0}, Φ_{0}, Ψ, Φ_{0})
= Q(Φ_{0}, Φ_{0}, Φ_{0}, Ψ) =< Ψ, Φ_{0} >_{H} .

(Qii) The factorisation property

Q(Φ_{X_{1}}^{K_{1}}, .., Φ_{X_{4}}^{K_{4}}) = Q(Φ_{∪_{i=2,..,4}X_{i}\X_{1}}^{K_{1}}, .., Φ_{X_{4}}^{K_{4}}) Q(Φ_{X_{1}\∪_{i=2,..,4}X_{i}}^{K_{1}}, Φ_{0}, Φ_{0}, Φ_{0}) . (Qiii) There is a constant C_{0} in (0, ∞) such that

sup_{X_{1},K_{1},..,X_{4},K_{4}} |Q(Φ_{X_{1}}^{K_{1}}, .., Φ_{X_{4}}^{K_{4}})| ≤ C_{0}^{N} ,

with N ≡ max_{l} |X_{l}|. In this setting, we have the following property.

Theorem IV.2 (Gaussian bounds) There exists a constant C in (0, ∞) such that for any integer N and f_{N} in H_{N},

∥f_{N}∥_{4} ≤ C^{N}∥f_{N}∥_{2}.

Proof : Let f_{N} be in H_{N}, that is

f_{N} = ∑_{(K,X)} b_{K,X} Φ_{X}^{K}.

with the summation over X of cardinality |X| = N. Then

∥f_{N}∥_{2}^{2} = ∑_{(K,X)} \overline{b}_{K,X} b_{K,X} = ∑_{(K,X)} |b_{K,X}|^{2},

where \overline{b} is the complex conjugate of b, and we have

∥f_{N}∥_{4}^{4} = ∑_{K_{1},K_{2},K_{3},K_{4}} ∑_{X_{1},X_{2},X_{3},X_{4}} \overline{b}_{K_{1},X_{1}}b_{K_{2},X_{2}} \overline{b}_{K_{3},X_{3}}b_{K_{4},X_{4}}
Q(Φ_{X_{1}}^{K_{1}}, Φ_{X_{2}}^{K_{2}}, Φ_{X_{3}}^{K_{3}}, Φ_{X_{4}}^{K_{4}}).

To analyse more closely this expression, we note that by our assumption (Qiii), we have

|Q(Φ_{X_{1}}^{K_{1}}, Φ_{X_{2}}^{K_{2}}, Φ_{X_{3}}^{K_{3}}, Φ_{X_{4}}^{K_{4}})| ≤ C_{0}^{N}.

Moreover , using the factorization property (Qii) and the fact that < ϕ_{i}^{k}, ϕ_{i}^{0}, >_{H_{i}}= 0, for k ≠ 0 , we observe that the configurations {X_{1}, …, X_{4}} with only one particle at a given site, (that is such that for some l there is an i ∈ X_{l} \ ∪_{k≠l}X_{k}), do not contribute. Therefore one has to consider the configurations such that at any occupied site there are 2, 3 or 4 particles, that is for a given i ∈ ∪_{l=1,..,4}X_{l} we have #{l : i ∈ X_{l}} ≥ 2. Taking this into account, we can restrict the summation to the following sets X_{i} = ∪_{j≠i}X_{i,j}, where X_{i,j} = X_{i} ∩ X_{j}. Using these two observations, we have

∥f_{N}∥_{4}^{4} ≤ C_{0}^{N} ∑_{K_{1},K_{2},K_{3},K_{4}} ∑_{X_{i,j}} |b_{K_{1},X_{1,2}∪X_{1,3}∪X_{1,4}}| |b_{K_{2},X_{1,2}∪X_{2,3}∪X_{2,4}}| ·
|b_{K_{3},X_{1,3}∪X_{2,3}∪X_{3,4}}| |b_{K_{4},X_{1,4}∪X_{2,4}∪X_{3,4}}|.

Applying Cauchy-Schwarz inequality to the sum over X_{3,4}, one gets

∥f_{N}∥_{4}^{4} ≤ C_{0}^{N}  ∑_{K_{1},K_{2},K_{3},K_{4}} ∑_{X_{i,j};{i,j}≠{3,4}} |b_{K_{1},X_{1,2}∪X_{1,3}∪X_{1,4}}| |b_{K_{2},X_{1,2}∪X_{2,3}∪X_{2,4}}|

(∑_{X_{3,4}} |b_{K_{3},X_{1,3}∪X_{2,3}∪X_{3,4}}|^{2})^{1/2} (∑_{X_{3,4}} |b_{K_{4},X_{1,4}∪X_{2,4}∪X_{3,4}}|^{2})^{1/2} .

Iterative use of similar arguments yields

∥f_{N}∥_{4}^{4} ≤ C_{0}^{N} ∑_{K_{1},K_{2},K_{3},K_{4}}

( ∑_{X_{1,2},X_{1,3},X_{1,4}} |b_{K_{1},X_{1,2}∪X_{1,3}∪X_{1,4}}|^{2})^{\frac{1}{2}} ( ∑_{X_{1,2},X_{2,3},X_{2,4}} |b_{K_{2},X_{1,2}∪X_{2,3}∪X_{2,4}}|^{2})^{\frac{1}{2}}

( ∑_{X_{1,3},X_{2,3},X_{3,4}} |b_{K_{3},X_{1,3}∪X_{3,2}∪X_{3,4}}|^{2})^{\frac{1}{2}} ( ∑_{X_{1,4},X_{2,4},X_{3,4}} |b_{K_{4},X_{1,4}∪X_{2,4}∪X_{3,4}}|^{2})^{\frac{1}{2}}

Taking into the account that for a given set X_{1} there are at most 2^{3N} subsets X_{1,2}, X_{1,3}, X_{1,4}, one obtains

∥f_{N}∥_{4}^{4} ≤ C_{0}^{N}2^{6N} _{∑K_{1},K_{2},K_{3},K_{4}} (∑_{X_{1}} |b_{K_{1},X_{1}}|^{2})^{1/2} (∑_{X_{2}} |b_{K_{2},X_{2}}|^{2})^{1/2}

(∑_{X_{3}} |b_{K_{3},X_{3}}|^{2})^{1/2} (∑_{X_{4}} |b_{K_{4},X_{4}}|^{2})^{1/2} ,

that is we have

∥f_{N}∥_{4}^{4} ≤ 2^{6N}C_{0}^{N} [∑_{K} (∑_{X} |b_{K,X}|^{2})^{1/2}]^{4} .

The number of sets K is finite and bounded by D^{N}, where D ≡ dimH_{0}, therefore one can use Cauchy-Schwarz inequality to get

∥f_{N}∥_{4}^{4} ≤ 2^{6N}C_{0}^{N}D^{2N} [∑_{K} ∑_{X} |b_{K,X}|^{2}]^{2} = 2^{6N}C_{0}^{N}D^{2N} ∥f_{N}^{2}∥_{2}^{4}.

This concludes the proof of Theorem IV.2. ♢

In the next subsection we will show that the hypotheses of Theorem IV.1 are true in noncommutative product spaces. Since in these cases the hypotheses on the spectral properties of the dynamics are straightforward to check, to prove hypercontractivity it is enough to show that the conditions (Qi)-(Qiii) hold.

Hypercontractivity in Non-commutative Product Spaces

Let A be a C^{∗} - algebra defined as the inductive limit over a finite dimensional algebra M of complex matrices of dimension n. For A = {a_{l,m}}_{l,m=1,..,n} ∈ M, we define the normalised trace by

tr(A) = \frac{1}{n} ∑_{l=1}^{n} a_{l,l}.

Given a positive matrix ρ_{0} such that tr(ρ_{0}) = 1, we introduce a state ν_{0}(A) ≡ tr(ρ_{0}A) and for A, B ∈ M the associated scalar product ⟨· , · ⟩_{s} , s ∈ [0, 1] , given by

⟨ A , B ⟩_{s} = tr(ρ_{0}^{1−s}A^{∗}ρ_{0}^{s}B)

where A^{∗} is the adjoint of A in A. Let {ϕ^{k}}_{0≤k≤n^{2}−1} be an orthonormal basis of M for the above scalar product. We assume that ϕ^{0} = 1, where 1 is the identity matrix.

Given a finite subset X of S, we introduce a sub-algebra A_{X} in A isomorphic to M^{X}. For a given i ∈ S, by tr_{i} , ν_{i} and ⟨· , · ⟩_{i} we denote the isomorphic copy of tr , ν_{0} and the corresponding scalar product (with a fixed parameter s ) on A_{i} , respectively. Let {ϕ_{i}^{k}}_{0≤k≤n^{2}−1} be the orthonormal basis in A_{i} , i.e. we have

∀k, k′ ∈ {0, …, n^{2} − 1}, ⟨ (ϕ_{i}^{k})^{∗}ϕ_{i}^{k′}⟩_{i} = δ_{k,k′}.

A general simple local element of A can be formally represented by

Φ_{X}^{K} = ⊗_{i∈X}ϕ_{i}^{k(i)} ⊗_{j∈S/X} 1_{j},

where k(i) belongs to {1, …, n^{2} − 1}. The product of two simple elements is defined as follows

Φ_{X}^{K}Φ_{X′}^{K′} = ( ⊗_{i∈X} ϕ_{i}^{k(i)} ⊗_{j∈S/X} 1_{j}) ( ⊗_{i∈X′} ϕ_{i}^{k′(i)} ⊗_{j∈S/X′} 1_{j}),
= ⊗_{i∈X∩X′} ϕ_{i}^{k(i)}ϕ_{i}^{k′(i)} ⊗_{i∈X\X′} ϕ_{i}^{k(i)} ⊗_{i∈X′\X} ϕ_{i}^{k′(i)} ⊗_{j∈S/X′∪X} 1_{j}

By linearity it can be extended to arbitrary elements of A .

Let ω ≡ ⊗_{i∈S} ν_{i}. In A , for any integer p, we introduce 𝕃_{p}(ω) norm ∥.∥_{p} by

∀f ∈ A, ∥f∥_{p} = lim_{X⟶S} (Tr|ρ_{X}^{s/p}fρ_{X}^{(1−s)/p}|^{p})^{\frac{1}{p}} ,

where |F | = (F^{∗}F)^{\frac{1}{2}} and ρ_{X} = ⊗_{i∈X}ρ_{i} . Let H_{N} denote the 𝕃_{2}(ω) closure of Span{Φ_{X}^{K} : |X| = N}.

Exercise : Prove that for N_{1} ≠ N_{2} one has H_{N_{1}} ⊥ H_{N_{2}}(the corresponding spaces are orthogonal).

∘

In this setting we have the following Gaussian bounds.

Theorem IV.3 There exists a constant C ∈ (0, ∞) independent of N such that for any f_{N} in H_{N},

∥f_{N}∥_{4} ≤ C^{N}∥f_{N}∥_{2}.

Proof : We begin from noting that in the present setting we have

Q(Φ_{1}, Φ_{2}, Φ_{3}, Φ_{4}) = Tr(ρ_{X}^{(1−s)/4} Φ_{1}^{∗} ρ_{X}^{s/2} Φ_{2} ρ_{X}^{(1−s)/2} Φ_{3}^{∗} ρ_{X}^{s/2} Φ_{4} ρ_{X}^{(1−s)/4})

with the right hand side independent of X for any local Φ_{1}, Φ_{2}, Φ_{3}, Φ_{4} ∈ A_{X}. For later purposes we will need the following simple property.

Exercise : Check that if ρ_{0} is the identity matrix, the corresponding quartic form satisfies all the properties (Qi) - (Qiii).

Using this we check the following properties of Q

(I) If Ψ ∈ A_{X} , then

Q(Ψ, Φ_{0}, Φ_{0}, Φ_{0}) = Tr(ρ_{X}^{(1−s)/4} Ψ^{∗} ρ_{X}^{s/2} Φ_{0} ρ_{X}^{(1−s)/2} Φ_{0} ρ_{X}^{s/2} Φ_{0} ρ_{X}^{(1−s)/4})
= Tr(ρ_{X} Ψ^{∗}) =< Ψ, Φ_{0} >_{H}

and so we have

Q(Ψ, Φ_{0}, Φ_{0}, Φ_{0}) = Q(Φ_{0}, Ψ, Φ_{0}, Φ_{0})^{∗} = Q(Φ_{0}, Φ_{0}, Ψ, Φ_{0})
= Q(Φ_{0}, Φ_{0}, Φ_{0}, Ψ)^{∗} =< Ψ, Φ_{0} >_{H} .

(II) The factorisation property : Suppose X_{1} \ ∪_{i=2,..,4}X_{i} ≠ ∅. Then, with X = ∪_{i=1,..,4}X_{i}, we have

Q(Φ_{X_{1}}^{K_{1}}, .., Φ_{X_{4}}^{K_{4}}) =

= Tr(ρ_{X}^{(1−s)/4} (Φ_{X_{1}}^{K_{1}})^{∗} ρ_{X}^{s/2} Φ_{X_{2}}^{K_{2}} ρ_{X}^{(1−s)/2} (Φ_{X_{3}}^{K_{3}})^{∗} ρ_{X}^{s/2} Φ_{X_{4}}^{K_{4}} ρ_{X}^{(1−s)/4}) =

Tr(ρ_{X}^{(1−s)/4}(Φ_{∪_{i=2,..,4}X_{i}\X_{1}}^{K_{1}})^{∗}(Φ_{X_{1}\∪_{i=2,..,4}X_{i}}^{K_{1}})^{∗}ρ_{X}^{s/2}Φ_{X_{2}}^{K_{2}}ρ_{X}^{(1−s)/2}(Φ_{X_{3}}^{K_{3}})^{∗}ρ_{X}^{s/2}Φ_{X_{4}}^{K_{4}}ρ_{X}^{(1−s)/4})

Since ρ = ρ_{X_{1}\∪_{i=2,..,4}X_{i}} ρ _{∪_{i=2,..,4}X_{i}\X_{1}} and by our definition Φ_{X_{1}\∪_{i=2,..,4}X_{i}}^{K_{1}} and ρ_{X_{1}\∪_{i=2,..,4}X_{i}} commutes with Φ_{∪_{i=2,..,4}X_{i}\X_{1}}^{K_{1}}, Φ_{X_{m}}^{K_{m}} , m = 2, 3, 4 , and ρ ∪_{i=2,..,4}X_{i}\X_{1} , using the factorisation property for the trace we arrive at

Q(Φ_{X_{1}}^{K_{1}}, .., Φ_{X_{4}}^{K_{4}}) = Q(Φ_{∪_{i=2,..,4}X_{i}\X_{1}}^{K_{1}}, .., Φ_{X_{4}}^{K_{4}}) Q(Φ_{X_{1}\∪_{i=2,..,4}X_{i}}^{K_{1}}, Φ_{0}, Φ_{0}, Φ_{0}) .

(III) There is a constant C_{0} in (0, ∞) such that

sup_{X_{1},K_{1},..,X_{4},K_{4}} Q(Φ_{X_{1}}^{K_{1}}, .., Φ_{X_{4}}^{K_{4}}) ≤ C_{0}^{N} ,

with N ≡ max_{l} |X_{l}|.

Given these properties, Theorem IV.2 implies the desired result.

♢

Next we introduce a densely defined Markov generator L as follows

Lf ≡ ∑_{i∈S}(ν_{i}f − f),

with ν_{i} denoting the partial state (conditional expectation) with respect to A_{S\i} .Since by our definition of the basis in H_{N} we have for any finite set X of size |X| = N ∈ ℕ and a function K

ν_{i}Φ_{X}^{K} = \{ \begin{matrix}0 & if & i ∈X \\ Φ_{X}^{K} & & otherwise \end{matrix}

one can see that

LΦ_{X}^{K} ≡ ∑_{i∈S}(ν_{i}Φ_{X}^{K} − Φ_{X}^{K}) = −N · Φ_{X}^{K}

Hence, we get that the corresponding semigroup P_{t} ≡ e^{tL} preserves the Nparticle subspaces H_{N}, for any N ∈ ℕ and that the spectrum of the restriction L|H_{N} equals to −N. Thus, using Theorem IV.1, we conclude the proof hypercontractivity of the free semigroup in 𝕃_{p}(ω) spaces.

Theorem IV.4 The (Markov - Feller) semigroup P_{t} ≡ e^{tL} on A with generator defined by

Lf ≡ ∑_{i∈S}(ν_{i}f − f),

is hypercontractive in the non-commutative 𝕃_{p}(ω) spaces.

To extend this result further, we consider the symmetric spaces with parameter s = \frac{1}{2} and recall that in this situation we have introduced quantum entropy QE_{p} , p ∈ [1, ∞) , formally by

QE_{p}(f) ≡ lim_{Λ→S} Tr |ρ_{Λ}^{1/2p}fρ_{Λ}^{1/2p}|^{p} (log |ρ_{Λ}^{1/2p}fρ_{Λ}^{1/2p}| − 1/p log ρ_{Λ}) . (130)

Exercise : Check that for a product state ω the relative entropy QE_{p}(f) is well defined on a dense set of local elements f ∈ A and can be extended to the space A_{exp} ≡ ⊕_{N}e^{εN} ·H_{N} with some sufficiently large ε

∘

Using this quantum entropy we have

Theorem IV.5 The hypercontractivity of the semigroup P_{t} ≡ e^{tL} in the non-commutative 𝕃_{p}(ω) spaces is equivalent to the following Logarithmic Sobolev inequality

QE_{2}(f) ≦ c < L^{(X)}f , f >_{𝕃_{2}(ω)}

We remark that, if S = ℤ^{d}, the following more general result can be proven

Theorem IV.6 For any finite set X ⊂ ℤ^{d} the Markov - Feller semigroup on A with generator defined by

L^{(X)}f ≡ ∑_{j∈ℤ^{d}}(ν_{X+j}f − f),

with ν_{X+j} being a partial state with respect to A_{S\X+j} where X + j denotes the translation of a given finite set X ⊂ ℤ^{d} by a vector j ∈ ℤ^{d}, is hypercontractive in the non-commutative 𝕃_{p}(ω) spaces. Equivalently the following Logarithmic Sobolev inequality is satisfied

QE_{2}(f) ≦ c < L^{(X)}f , f >_{𝕃_{2}(ω)}

To prove this result it is sufficient to note that, by results of [36] described in Lecture III, the hypercontractivity for the semigroup considered before is equivalent to a Logarithmic Sobolev inequality and then we notice that the Dirichlet forms corresponding to arbitrary X are equivalent to the quadratic form of single spin flip generator described above (as shown in Appendix I) and 𝕃_{p} regular (as described in Lecture III).

An interesting new feature of the general case is a more complicated structure of the invariant subspaces and the spectrum. In particular the spectrum of the restriction of the generator to consecutive subspaces lies in overlapping intervals.

Following Lecture III one can show the corresponding Logarithmic Sobolev inequality with a quantum relative entropy given by (130). Note that, (by exercise above after the formula for entropy), the limit on the right hand side exists for sufficiently large set of observables. By similar arguments as in Appendix I, one can show that the Dirichlet form of L is equivalent (in 𝕃_{2}(ω)) to the one of the generator L^{(0)} defined with the generalized expectations E_{{i}}, i ∈ ℤ^{d}. Hence we get the following result.

Theorem IV.7 : There exists a constant c_{Λ_{0}} in (0, ∞) such that the following Logarithmic Sobolev inequality is true,

QE_{2}(f) ≤ c_{Λ_{0}} ⟨f, −L^{(0)}f⟩_{𝕃_{2}(ω, \frac{1}{2} )}

By this we have got a weak form of the product property of the Logarithmic Sobolev inequality in the non-commutative setting, (that is we have proved Logarithmic Sobolev inequality for the product space, but possibly with a non-optimal constant) .

Appendix I

Generalised Conditional Expectations and Elementary Markov operators

For the general theory of generalised conditional expectations in a von Neuman algebras we refer to [2]. As discussed in the Lecture I in order to construct and study ergodicity properties of nontrivial infinite dimensional Markov semigroups it is vital to have an explicit representation of them allowing a proof of certain regularity estimates. In this Appendix we provide more examples of generalised conditional expectations in a context of von Neuman algebras.

We begin from the conditional expectations relevant to extension of the Glauber type dynamics. Let ω ≡ ω_{βΦ} be a Gibbs state corresponding to an interaction Φ ∈ 𝔹_{exp} , (that is satisfying ||Φ||_{exp} ≡ sup_{i∈ℤ^{d}} ∑_{X⊂⊂ℤ^{d}, X∋i} e^{ε|X|}||Φ|| < ∞, for some ε > 0), at some inverse temperature β. Let π_{ω} denotes the GNS representation associated to a Gibbs state ω.

We are interested in constructing a generalised conditional expectation in the von Neumann algebra M ≡ π_{ω}(A)′′ with respect to the algebra generated by π_{ω}(τ_{X}(A)) , the representation of τ_{X}(A) where τ_{X} is a perfect conditional expectation in A such that we have

τ_{X}(f) = f for any f ∈ A_{X^{c}}

The example of such conditional expectation given by the partial trace Tr_{X} was (discussed in Lecture I and) given by

Tr_{X}(f) ≡ ∫ ν_{X}(dU)U^{∗}fU,

with ν_{X}(dU) denoting the (normalised) Haar measure on the unitary group of A_{X} . This formula is particularly convenient for passing to a representation. By choosing to integrate with a normalised Haar measure over some other subgroup we can obtain different examples. One particular example is given by using a local automorphism σ_{X} in the inductive limit algebra A, which satisfies

(i) σ_{X}(f) = f for any f ∈ A_{X^{c}}

and

(ii) σ_{X}^{2} = 1

We assume also that the automorphism σ_{X} preserves the partial trace Tr_{X}, i.e. we have

(iii) Tr_{X} (σ_{X}(f)) = Tr_{X}(f)

We observe that our local automorphism induces an automorphism, (denoted later by the same symbol), on the algebra M. (This is because the automorphism σ_{X}, as an automorphism acting effectively only on a finite dimensional algebra A_{X}, is inner.)

Let τ_{X} be a projection on M defined as follows

τ_{X}(f) ≡ \frac{1}{2} (1 + σ_{X}) (f)

for any f ∈ M. Setting φ_{1} to be the state defined by the (cyclic and separating) vector Ω_{ω}, we introduce another state φ_{2} by the following formula

φ_{2}(f) ≡ φ_{1} (τ_{X}(f))

In [28] the following result was shown.

Proposition : Suppose |β| < 2β_{0}, where β_{0} ∈ (0, ∞) is the radius of analyticity for a potential Φ ∈ 𝔹_{exp}, or in one dimension that the interaction Φ is of finite range. Then for any finite set X ⊂⊂ ℤ^{d} there is a constant λ ∈ (0, ∞) such that for any f ∈ π_{1}(A^{+}) we have

λφ_{1}(f) ≤ φ_{2}(f) ≤ λφ_{1}(f)

and therefore the state φ_{2} is a faithful on the von Neumann algebra M.

Thus, by a result of Connes [14],[15], the map t ↦ (Dφ_{1} : Dφ_{2})_{t} has an analytic continuation in t to the strip −\frac{1}{2} ≤ Imz ≤ \frac{1}{2} , (with respect to the scaled by β time), and

η_{X} ≡ (Dφ_{1} : Dφ_{2})_{|t=−i/2}

is ă bounded operator belonging to M. Using that we define a generalized conditional expectation 𝔼_{X} in the von Neumann algebra M by the following formula

𝔼_{X}(f) ≡ Tr_{X} (η_{X}^{∗}fη_{X})

It is interesting to observe that the following is true.

(i) By construction of φ_{2}, the partial trace τ_{X} is symmetric in 𝕃_{2}(φ_{2}, \frac{1}{2})

< τ_{X} f, g >_{φ_{2}, \frac{1}{2}} =< f, τ_{X} g >_{φ_{2}, \frac{1}{2}}

(ii) One has the following relation between scalar products in 𝕃_{2}(φ_{2}, \frac{1}{2}) and 𝕃_{2}(φ_{1}, \frac{1}{2})

< f, g >_{𝕃_{2}(φ_{1}, \frac{1}{2} )}=< η_{X}^{∗}fη_{X}, g >_{𝕃_{2}(φ_{2}, \frac{1}{2} )}=< f, η_{X}^{∗}gη_{X} >_{𝕃_{2}(φ_{2}, \frac{1}{2} )}

Then the symmetry of 𝔼_{X}(·) comes out from the following sequence of equalities, in which < ·, · >_{i} denotes the scalar product in 𝕃_{2}(φ_{i}, \frac{1}{2}), i = 1, 2, respectively,

< 𝔼_{X}(f), g >_{1}=< 𝔼_{X}(f), η_{X}^{∗}gη_{X} >_{2}=< τ_{X}(η_{X}^{∗}fη_{X}), η_{X}^{∗}gη_{X} >_{2}
= < (η_{X}^{∗}fη_{X}), τ_{X}(η_{X}^{∗}gη_{X}) >_{2}=< (η_{X}^{∗}fη_{X}), 𝔼_{X}(g) >_{2}
= < f, 𝔼_{X}(g) >_{1}

Using the generalized conditional expectations we can introduce elementary Markov generators of jump type on the algebra M by, (cf. [28]),

L_{X+j}f ≡ 𝔼_{X+j}(f) − f

where X ⊂⊂ ℤ^{d}, j ∈ ℤ^{d} and f ∈ M. Thus we have got an elementary Markov generator for infinite volume dynamics, which has essentially the same form as in the finite volume case where a state is given by a density matrix. By the construction, L_{X} maps M into M and not only preserves the state φ_{1} , but is also symmetric in an appropriate Hilbert space .

Examples .

The following explicit example of the generalised expectation is useful for defining the quantum analogue of Kawasaki dynamics.

(E1) Quantum Exchange

For X = {i, j}, with some i, j ∈ ℤ^{d}, we define the following automorphism in A

σ_{X} ≡ ι_{X^{c}} ⊗ e_{ij}

where for Y ⊂ ℤ^{d} we have used ι_{Y} to denote the unit automorphism on A_{Y} , and e_{ij} is an exchange automorphism in A_{X} defined on the basis {a_{i}^{k} ⊗ a_{j}^{l} : k, l = 1, .., dimM}, as follows

e_{ij}(a_{i}^{k} ⊗ a_{j}^{l}) ≡ a_{i}^{l} ⊗ a_{j}^{k}

It is not difficult to see that the automorphism e_{ij} satisfies

e_{ij}^{2} = ι_{{ij}}

and

Tr_{ij} (e_{ij}(f)) = Tr_{ij}(f)

Hence the corresponding automorphism σ_{X} satisfies all the desired properties (i) - (iii). Thus using the construction presented in the above we can define the corresponding generalized conditional expectation Λ_{ij} and introduce the following Markov generator in the algebra M

D_{ij}(f) ≡ Λ_{ij}(f) − f

We call the corresponding stochastic dynamics a quantum exchange stochastic dynamics.

(E2) A Stochastic Dynamics with Partial Decoupling.

Choosing a local partition of a unit, i.e. a family of projectors Q ≡ {q_{n} ∈ A_{X} : n ≤ dimA_{X}} such that q_{n}^{∗} = q_{n}, q_{m}q_{n} = δ_{mn}q_{n}, and ∑_{n} q_{n} = 1, and defining a map

A ∋ f ⟼ Q(f) ≡ ∑_{n} q_{n}fq_{n}

By this definition the map Q is a projection (i.e. Q^{2} = Q) of A onto Q(A). It is also not difficult to see that

Tr_{X} (Q(f)) = Tr_{X}(f)

This formula induces a map, denoted later by the symbol Q, in the von Neumann algebra M ≡ π_{ω}(A)′′ associated to a Gibbs state ω. The map Q has all necessary properties for our strategy of constructing the generalized conditional expectations .

𝔼_{Q}(f) ≡ Q (ζ_{Q}^{∗}fζ_{Q})

with suitable ζ_{Q} ∈ M and we define a Markov generator

L_{Q}(f) = 𝔼_{Q}(f) − f

Quadratic Forms

It is known that in the case of commutative spin systems of classical statistical mechanics one can get ergodicity for the larger range of temperatures for stochastic dynamics defined with the elementary generators L_{X} associated with translations of a cube X of large size, see [3]. In commutative situation there is a standard way to use this the strong ergodicity result for dynamics of block spin flips type with some large block X , to prove the 𝕃_{2} ergodicity of all the other block spin flip generators. The main idea is based on the equivalence of Dirichlet forms associated to generators L^{Y} with Y ⊂⊂ ℤ^{d}. We will show below that this equivalence remains true in the considered case of quantum block spin flip generators associated to a finite range classical potential Φ and we have the following result

Theorem AI.1 : For every X, Y ⊂⊂ ℤ^{d} there is a constant C_{X,Y} ∈ (0, ∞) such that

C_{X,Y}^{−1} ⟨f, −L_{Φ}^{X}f⟩_{𝕃_{2}(ω_{Φ}, \frac{1}{2} )} ≤ ⟨f, −L_{Φ}^{Y}f⟩_{𝕃_{2}(ω_{Φ}, \frac{1}{2} )} ≤ C_{X,Y} ⟨f, −L_{Φ}^{X}f⟩_{𝕃_{2}(ω_{Φ}, \frac{1}{2} )}

Hence, if for some X ⊂⊂ ℤ^{d} the corresponding dynamics e^{tL_{Φ}^{X}} is strongly exponentially ergodic, then for any Y ⊂⊂ ℤ^{d} the corresponding block spin flip generator L_{Φ}^{Y} as a selfadjoint operator in L_{2}(ω_{Φ}, \frac{1}{2}) has a spectral gap and the corresponding stochastic dynamics is 𝕃_{2} - ergodic.

Proof: It is sufficient to show the equivalence of the Dirichlet forms in case when X = {0}, that is the equivalence of any block spin flip generator with the single spin flip one. One can reduce our problem to proper estimates on elementary generators. We will need the following proposition in which we use the following notation

∂_{R}Y = {k ∈ ℤ^{d} \ Y : ∀j ∈ Y |k − j| ≤ R}.

To simplify the notation later on we will set ⟨·, ·⟩ ≡ ⟨·, ·⟩_{𝕃_{2}(ω_{Φ}, \frac{1}{2} )} and || · ||_{2} ≡ || · ||_{𝕃_{2}(ω_{Φ}, \frac{1}{2} )}.

Proposition AI.2 : Let Φ be a classical potential of finite range R. Then for Z ≡ Y ∪ ∂_{R}Y, we have

⟨f, −L_{Y} f⟩ ≤ C_{Z} ∑_{i∈Z} ∥∂_{i}f∥_{2}^{2}  (131)

where C_{Z} is a positive constant depending on Z while ∂_{i}f ≡ f − T r_{{i}}f . Moreover, there is a constant C > 0 such that for any i ∈ ℤ^{d} we have

∥∂_{i}f∥_{2}^{2} ≤ C⟨f, −L_{{i}}f⟩  (132)

Applying this proposition to L_{Y +j}, by summation over j it is easy to get the inequality

⟨f, −L^{Y} f⟩ ≤ C_{Y,0}⟨f, −L^{0}f⟩

Proof of Proposition AI.2 : Let us first observe that for Y ∪ ∂_{R}Y ⊂ Z we have

E_{Y} Tr_{Z}f = Tr_{Y} γ_{Y}^{∗} Tr_{Z}fγ_{Y} = Tr_{Z}fTr_{Y} γ_{Y}^{∗} γ_{Y} = Tr_{Z}f  (133)

where we have used that γ_{Y} = e^{− \frac{1}{2} U_{Y}} (Tr_{Y} e^{−U_{Y}} )^{− \frac{1}{2}} commutes with any operator localized in ℤ^{d} \ {Y ∪ ∂_{R}Y }. Hence L_{Y} Tr_{Z}f = 0 and we have

⟨f, −L_{Y} f⟩ = ⟨f, −L_{Y} (f − Tr_{Z}f)⟩ = ⟨−L_{Y} f, (f − Tr_{Z}f)⟩ (134)

≤ \frac{a}{2}|| − L_{Y} f||_{2}^{2} + \frac{1}{2a}||(f − Tr_{Z}f)||_{2}^{2} ≤ \frac{a}{2}∥L_{Y} ∥ · ⟨f, −L_{Y} f⟩ + \frac{1}{2a}||(f − Tr_{Z}f)||_{2}^{2}

for any a ∈ (0, ∞); we have used here the fact L_{Y} is a symmetric operator on the Hilbert space 𝕃_{2} with a bounded norm ∥L_{Y} ∥ ≡ ∥L_{Y} ∥_{𝕃_{2}→𝕃_{2}} (cf. [28]). Choosing

\frac{a}{2}∥L_{Y} ∥ < 1  (135)

we arrive at

⟨f, −L_{Y} f⟩ ≤ C_{1}||(f − Tr_{Z}f)||_{2}^{2}  (136)

with C_{1} ≡ a^{−1} · (2 − a∥L_{Y} ∥)^{−1}. Now we observe

∥f − Tr_{Z}f∥_{2}^{2} = |Z|^{2}∥\frac{1}{|Z|} ∑_{i_{n}∈Z}(Tr_{\{k<i_{n}\}}f − Tr_{\{k≤i_{n}\}}f)∥_{2}^{2}
≤ |Z| ∑_{i_{n}∈Z} ∥Tr_{{k<i_{n}}}f − Tr_{{k≤i_{n}}}f∥_{2}^{2}  (137)

where in the last step the Holder inequality was used; |Z| denotes the number of sites in Z and we assumed a convention f ≡ Tr_{{∅}}f. To complete the proof of the first part of the Proposition we need to observe the following fact

Lemma AI.3 : Under the above assumptions,

∥Tr_{{l}}f∥_{2} ≤ ∥Tr_{{l}}∥ · ∥f∥_{2}  (138)

Proof of Lemma AI.3 : It is sufficient to prove the bound of interest to us for finite volume states, because by appropriate limiting procedure the infinite volume bound follows. Let 𝕃_{2}(ω, s), s ∈ [0, 1], be the one parameter family of interpolating Hilbert spaces introduced in [27] (see also [28].) We recall that the scalar product of 𝕃_{2}(ω, s) is given by

< f, f >_{ω_{ϱ},s}= Tr(ϱ^{(1−s)}f^{∗}ϱ^{s}f)

(In particular, ∥ · ∥_{2} ≡ ∥ · ∥_{𝕃_{2}(ω, \frac{1}{2} )}.) We have

ω(|Tr_{{l}}f|^{2}) ≤ ω(Tr_{{l}}|f|^{2}) ≤ ∥Tr_{{l}}∥ · ω(|f|^{2})  (139)

and

ω(|Tr_{{l}}f^{∗}|^{2}) ≤ ω(Tr_{{l}}|f^{∗}|^{2}) ≤ ∥Tr_{{l}}∥ · ω(|f^{∗}|^{2})

where the Kadison-Schwarz inequality for Tr_{{l}}(·) was used. Then (138) follows from the interpolation procedure based on the Three Lines Theorem.

Now applying Lemma AI.3 to the inequality (137) we get

∥f − Tr_{Z}f∥_{2}^{2} ≤ |Z| ∑_{i_{n}∈Z} ∥Tr_{{k<i_{n}}}∥^{2}∥f − Tr_{{i_{n}}}f∥_{2}^{2}  (140)

≤ |Z|∥Tr_{{l}}∥^{2|Z|} ∑_{i∈Z} ∥f − Tr_{{i}}f∥_{2}^{2} ≤ |Z|∥Tr_{{l}}∥^{2|Z|} ∑_{i∈Z} ∥∂_{{i}}f∥_{2}^{2}

Consequently, (140) and (136) lead to

⟨f, −L_{Y} f⟩ ≤ C_{1}|Z| · ∥Tr_{{l}}∥^{2|Z|} ∑_{i∈Z} ∥∂_{{i}}f∥_{2}^{2}

This ends the proof of the first part of Proposition AI.2 . The second part follows from the following lemma. Lemma AI.4 : Under the conditions stated above,

∥f − Tr_{{i}}f∥_{2}^{2} ≤ C_{2}⟨f, −L_{{i}}f⟩  (141)

where C_{2} is a positive constant. Proof: The proof is based on the following simple observation.

Tr_{{i}}E_{{i}}f = Tr_{{i}}(Tr_{{i}}γ_{{i}}^{∗}fγ_{{i}}) = Tr_{{i}}Tr_{{i}}γ_{{i}}^{∗}fγ_{{i}}

= Tr_{{i}}γ_{{i}}^{∗}fγ_{{i}} = E_{{i}}f  (142)

So

∂_{{i}}E_{{i}}f = 0  (143)

Therefore, using Lemma AI.3, we get

∥∂_{{i}}f∥_{2}^{2} = ∥∂_{{i}}(f − E_{{i}}f)∥_{2}^{2} ≡ ∥∂_{{i}}(L_{{i}}f)∥_{2}^{2} ≤
≤ (1 + ∥Tr_{{i}}∥)^{2} · ∥L_{{i}}f∥_{2}^{2} ≤ (1 + ∥Tr_{{i}}∥)^{2} · ∥L_{{i}}∥⟨f, −L_{{i}}f⟩

This ends the proof of Lemma AI.4 and so also the proof of Proposition AI.2.

To prove the lower bound of interest to us it is sufficient to prove the following result. Proposition AI.5 : Let Y be a finite region of the lattice ℤ^{d} and let {i} ⊂ Y. Then

∥∂_{{i}}f∥_{2}^{2} ≤ C_{3}⟨f, −L_{Y} f⟩  (144)

where C_{3} is a positive constant independent of i. Moreover we have

⟨f, −L_{{i}}f⟩ ≤ C_{4} ∑_{j∈{i}∪∂_{R}{i}}⟨f, −L_{Y +j}f⟩  (145)

for any Y ⊂⊂ ℤ^{d} ∋ 0 with some constant C_{4} ∈ (0, ∞) independent of i.

Proof : For any Y ⊂⊂ ℤ^{d} ∋ 0 we have

∥∂_{{j}}f∥_{2}^{2} = ∥∂_{{j}}(L_{Y +j}f)∥_{2}^{2} ≤ 1 + ∥Tr_{{j}}∥)^{2}∥L_{Y +j}f∥_{2}^{2}  (146)
≤ 1 + ∥Tr_{{j}}∥)^{2} · ∥L_{Y +j}∥ · ⟨f, −L_{Y} f⟩

This ends the proof of the first part of Proposition AI.5. Now applying first part of Proposition AI.2 with the one point set {i} and Z = {i} ∪ ∂_{R}{i}, we get

⟨f, −L_{{i}}f⟩ ≤ C_{Z} ∑_{j∈{i}∪∂_{R}{i}} ∥∂_{j}f∥_{2}^{2}

and by the first part of Proposition AI.2 we arrive at

⟨f, −L_{{i}}f⟩ ≤ C_{4} ∑_{j∈{i}∪∂_{R}{i}}⟨f, −L_{Y +j}f⟩

with some nonnegative constant C_{4}.

Summing inequality (145) over i ∈ ℤ^{d} we arrive at

⟨f, −L^{0}f⟩ ≤ C_{Y,0}⟨f, −L^{Y} f⟩  (147)

This together with the inequality after Proposition AI.2 prove the equivalence of Dirichlet forms of L^{Y} and L^{0} for any finite set Y ⊂⊂ ℤ^{d}. From this the general case easily follows.

By this we finish the proof of equivalence of Dirichlet forms of all block spin flip generators.

Appendix II

𝕃_{p} Regularity of Dirichlet Forms

In this Appendix we discus in more details the 𝕃_{p} regularity of Dirichlet forms which plays a key role in the proof of the integration lemma (Theorem 4.8 Lecture III). In particular we show that the generalised Dirichlet form of any bounded generator and the generator of any subordinated semigroup possess the 𝕃_{p} regularity. Additionally this property is also true for if the underlying state is a trace and the generator is symmetric in the corresponding 𝕃_{2} space.

Recall that the form E_{p} form is called 𝕃_{p} regular iff there is d_{0} ≥ 0 such that ∀ q ≥ 2 and g ∈ D(E_{q}) we have

E_{2}(I_{2,q}(g), I_{2,q}(g)) ≤ \frac{q^{2}}{4(q − 1)} E_{q}(g, g) + d_{0}∥g∥_{q}^{q}

where E_{q}(g, g) ≡ < ϕ_{p}(g), −L_{q}g > ·∥g∥_{𝕃_{q}}^{q−2} = < I_{p,q}(g), −L_{q}g > with \frac{1}{p} + \frac{1}{q} = 1. Note that replacing g by I_{q,2}(f) , (and using Theorem 2.5 of Lecture III), we can rewrite the regularity condition as follows

E_{2}(f, f) ≤ \frac{q^{2}}{4(q − 1)} E_{2}(I_{p,2}(f), I_{q,2}(f)) + d_{0}∥f∥_{2}^{2}  (148)

Clearly that is equivalent to the following inequality:

lim_{t→0} \frac{1}{t} < f, (1 − P_{t})f > ≤ \frac{q^{2}}{4(q − 1)} lim_{t→0} \frac{1}{t} < I_{p,2} (f), (1 − P_{t})I_{q,2} (f) >
+ d_{0}∥f∥_{2}^{2}

where f ∈ 𝕃_{2}^{+} ∩ D(L_{2}) is such that I_{q,2}(f) ∈ D(L_{q}) and \frac{1}{p} + \frac{1}{q} = 1. The last is certainly satisfied if for all small t ≥ 0

< f, (1 − P_{t})f >≤ \frac{q^{2}}{4(q − 1)}(< I_{p,2} (f), I_{q,2} (f) > − < I_{p,2} (f), P_{t}I_{q,2} (f) >)

+td_{0}∥f∥_{2}^{2}

But

< I_{p,2} (f), I_{q,2} (f) > = < I_{p,q}(I_{q,2} (f)), I_{q,2} (f) > = ∥I_{q,2} (f)∥_{q}^{q} = ∥f∥_{2}^{2} so we can rewrite the above inequality in the following form: < I_{p,2} (f), P_{t}I_{q,2} (f) > ≤ ξ_{q} < f, P_{t}f > + (1 − ξ_{q})∥f∥_{2}^{2} + ξ_{q}td_{0}∥f∥_{2}^{2} (149)

where ξ_{q} = 4(q − 1)/q^{2}. Define the following functions

h(q) = < I_{p,2} (f), P_{t}I_{q,2} (f) >

for q ∈ (1, ∞), and

h_{0}(s) = h(\frac{2}{s})

where s ∈ (0, 2). It is clear that they are positive. Two other simple properties of the function h_{0} are contained in the following lemma. Lemma AII.1 The function h_{0} is symmetric around the point s = 1, that is we have h_{0}(1 + s) = h_{0}(1 − s) if s ∈ [0, 1), and h_{0}(s) ≤ ∥f∥_{2}^{2} for every s ∈ (0, 2). Proof : We have h_{0}(1 + s) =< I_{\frac{2}{1−s} ,2} (f), P_{t}I_{\frac{2}{1+s} ,2} (f) >=< I_{\frac{2}{1+s} ,2} (f), P_{t}I_{\frac{2}{1−s} ,2} (f) >= h_{0}(1 − s) The second statement follows from the Hölder inequality: h_{0}(s) =< I_{\frac{2}{2−s} ,2} (f), P_{t}I_{\frac{2}{s} ,2} (f) > ≤ ∥I_{\frac{2}{2−s} ,2} (f)∥_{\frac{2}{2−s}} ∥P_{t}I_{\frac{2}{s} ,2} (f)∥_{\frac{2}{s}} ≤ ∥f∥_{2}^{2} □

One can express the inequality (149) in terms of the function h_{0} as follows.

h_{0}(s) ≤ (∥f∥_{2}^{2} − h_{0}(1))(s − 1)^{2} + h_{0}(1) + td_{0}(2 − s)s∥f∥_{2}^{2} (150)

Note that for s ∈ (1, 2) we have h_{0}(s) ≤ ∥f∥_{2}^{2}, thus the condition (149) is certainly fulfilled if we have

(∥f∥_{2}^{2} − h_{0}(1))(s − 1 )^{2} + h_{0}(1) + td_{0}(2 − s)s∥f∥_{2}^{2} − ∥f∥_{2}^{2} ≥ 0 But that is satisfied iff

∥f∥_{2}^{2} − h_{0}(1) ≤ td_{0}∥f∥_{2}^{2} ⇔ \frac{< f, (1 − P_{t})f >}{t∥f∥_{2}^{2}} ≤ d_{0}

The last is naturally true for small t provided we choose d_{0} = (∥L_{2}∥ + 1). Hence we conclude that.

Theorem AII.2. (Bounded Generators have Regular Forms) If the generator L_{2} is bounded, then (150) holds with d_{0} = (∥L_{2}∥ + 1).

To study the properties of the function h_{0} the following notion will prove to be useful.

Definition AII.3 A function g : [0, β] → R is Reflection Positive (RP) iff it is continuous and for every n ∈ ℕ, any z_{1}, ..., z_{n} ∈ ℂ and any s_{1}, ..., s_{n} ∈ [0, β/2] the following inequality holds:

∑_{i,j=1}^{n} \overline{z_{i}}z_{j}g(s_{i} + s_{j}) ≥ 0

We will need the following lemma.

Lemma AII.4 If the function h_{0} is RP on [0, 2], then (150) holds with d_{0} = 0.

Proof : We need to show that

h_{0}(s) ≤ (∥f∥_{2}^{2} − h_{0}(1))(s − 1)^{2} + h_{0}(1)

Because h_{0} is symmetric around s = 1, it is enough to consider s ∈ [1, 2). We have that h_{0}(s) ≤ ∥f∥_{2}^{2} and it is continuous on [0, 2] so h_{0}(2) ≤ ∥f∥_{2}^{2} too. Moreover from Widder theorem, [42], h_{0} possesses the following integral representation

h_{0}(s) := ∫_{−∞}^{∞} e^{−λs}dν(λ)

where ν is a finite Borel measure on ℝ. So h_{0}^{2n−1}(1) = 0 and h_{0}^{(2n)}(1) ≥ 0 for every natural n. Thus h_{0}(2) ≥ h_{0}(1) and so

h_{0}(s) = h_{0}(1) + (s − 1)^{2} ∑_{n=1}^{∞} h_{0}^{(2n)}(1)\frac{(s − 1)^{2n−2}}{(2n)!}

≤ h_{0}(1) + (s − 1)^{2} ∑_{n=1}^{∞} h_{0}^{(2n)}(1)\frac{1}{(2n)!} ≤ h_{0}(1) + (s − 1)^{2}(∥f∥_{2}^{2} − h_{0}(1)) □

Using the above lemma we prove the following result :

Theorem AII.5 (Regularity in 𝕃_{p}(Tr) spaces) Suppose P_{t} is a 𝕃_{2}-symmetric and Markov-Feller semigroup in 𝕃_{p}(Tr) spaces. Then the corresponding generalized Dirichlet forms are 𝕃_{p} - regular.

Proof : In 𝕃_{p}(ω) spaces, for any f ∈ A^{+}, one has

h_{0}(s) =Trρ^{\frac{s}{4}} (ρ^{\frac{1}{4}} fρ^{\frac{1}{4}} )^{2−s}ρ^{\frac{s}{4}} P_{t}(ρ^{− \frac{s}{4}} (ρ^{\frac{1}{4}} fρ^{\frac{1}{4}} )^{s}ρ^{− \frac{s}{4}} ),

where s ∈ [0, 2]. If ρ = 1, it simplifies to h_{0}(s) = Trf^{2−s}P_{t}(f^{s}). Since P_{t} is Markov and 𝕃_{2}(Tr)-symmetric, using the representation given in [4], we obtain

h_{0}(s) = ∫ ∫ dμ_{f}(ξ, β)ξ^{s}β^{2−s},

where dμ_{f} is a positive and symmetric measure with support contained in Spec(f)×Spec(f). It can be easily seen that this is a RP function. Hence inequality (150) holds with d_{0} = 0. By continuity it is satisfied for all f ∈ 𝕃_{q}^{+}(Tr):

< I_{2,q}(f), \frac{1 − P_{t}}{t} I_{2,q}(f) >_{Tr}≤ ξ_{q} < I_{p,q}(f), \frac{1 − P_{t}}{t} I_{p,q}(f) >_{Tr}

Let additionally f ∈ D(L_{q}). Then the limit t → 0 of the right hand side exists. The left hand side can be written as

∫_{0}^{∞} \frac{1 − e^{−tλ}}{t} < I_{2,q}(f), dE(λ)I_{2,q}(f) >_{Tr},

where dE(λ) is the spectral measure of −L_{2}, that is −L_{2} = ∫_{0}^{∞} λdE(λ). For t > 0 let us define η_{t}(λ) : [0, ∞) → R^{+}, by

η_{t}(λ) = \frac{1 − e^{−tλ}}{t}

Because for all λ ≥ 0 η_{t}(λ) ≥ η_{t′}(λ) if t < t′, the following limit exists

lim_{t→0} ∫_{0}^{∞}η_{t}(λ) < I_{2,q}(f), dE(λ)I_{2,q}(f) >_{Tr}

By the monotone convergence theorem lim_{t→0} η_{t}(λ) = λ is < I_{2,q}(f), dE(λ) I_{2,q}(f) >_{Tr} integrable. Thus I_{2,q}(f) ∈ D(\sqrt{−L_{2}}) = D(E_{2}) and

E_{2}(I_{2,q}(f), I_{2,q}(f)) ≤ ξ_{q}E_{q}(f, f) □

Generators of Subordinated Semigroups have 𝕃_{p} Regular Forms

Let P_{t} be a subordinated semigroup to the modular automorphism

α_{τ}(f) ≡ ρ^{−iτ}fρ^{iτ}

that is a semigroup given by

P_{t}(f) ≡ ∫ ν_{t}(dτ)α_{τ}(f)

with a measure ν_{t}(dτ) defined as the Fourier transform of exp{−tF (x)} , where F is given by the Lévy-Khinchine formula

F (x) = ax^{2} + ∫ (1 − cos xy) \frac{1 + y^{2}}{y^{2}} dμ(y)

where dμ is a positive and finite measure on R such that μ({0}) = 0. One can see that, for any z ∈ [0, 1], the semigroup P_{t} satisfies the following symmetry condition

< P_{t}f , g >_{ω,z} ≡ Tr (ρ^{1−z}P_{t}f^{∗}ρ^{z}g) = < f , P_{t}g >_{ω,z}

Since now for any p, q ∈ [1, ∞) we have

α_{τ}I_{p,q}(f) = I_{p,q}(α_{τ}f)

in the present situation

h_{0}(s) =Trρ^{\frac{s}{4}} (ρ^{\frac{1}{4}} fρ^{\frac{1}{4}} )^{2−s}ρ^{\frac{s}{4}} P_{t}(ρ^{− \frac{s}{4}} (ρ^{\frac{1}{4}} fρ^{\frac{1}{4}} )^{s}ρ^{− \frac{s}{4}} )
=Tr(ρ^{\frac{1}{4}} fρ^{\frac{1}{4}} )^{2−s}P_{t}(ρ^{− \frac{s}{4}} (ρ^{\frac{1}{4}} fρ^{\frac{1}{4}} )^{s}ρ^{− \frac{s}{4}} )

Therefore h_{0} satisfies the RP property and we have the following result

Theorem AII.6 The generalized Dirichlet forms associated to a subordinated Feller - Markov semigroup are 𝕃_{p} regular.

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