aa r X i v : . [ m a t h . OA ] D ec THE EMERGENCE OFNONCOMMUTATIVE POTENTIAL THEORY.
FABIO CIPRIANI
Abstract.
We review origins and developments of Noncommutative Potential theory asunderpinned by the notion of energy form. Recent and new applications are shown to ap-proximation properties of von Neumann algebras.
Contents
1. Introduction. 22. Commutative Potential theory 22.1. Classical Potential Theory 22.2. Beurling-Deny Potential Theory 33. Operator algebras 53.1. C ∗ -algebras as noncommutative topology 53.2. von Neumann Algebras as noncommutative measure theory 53.3. Weights, traces, states and the GNS representation 63.4. Morphisms of operator algebras 83.5. Positivity preserving and Markovian semigroups on operator algebras 94. Noncommutative Potential Theory 94.1. Dirichlet forms on C ∗ -algebras with trace d’apres Albeverio-Hoegh-Krohn 105. Dirichlet forms and Differential Calculus: bimodules and derivations 135.1. The derivation determined by a Dirichlet form 195.2. Decomposition of derivation, Beurling-Deny fromula revisited 205.3. Noncommutative potential theory and curvature in Riemannian Geometry 215.4. Voiculescu Dirichlet form in Free Probability 246. Dirichlet forms on standard forms of von Neumann algebras 256.1. Tomita-Takesaki Theory and Connes’ Radon-Nikodym Theorem 276.2. Symmetric embeddings 287. Application to approximation/rigidity propertiesof von Neumann algebras 317.1. Amenable groups 317.2. Bimodules and Connes correspondences 327.3. Amenable von Neumann algebras 337.4. Amenability and subexponential spectral growth rate of Dirichlet forms 347.5. Haagerup approximation property and discrete spectrum of Dirichlet forms 357.6. Property (Γ) and Poincar´e inequality for elementary Dirichlet forms 367.7. Property (T) 38 This work has been supported by Laboratoire Ypatia des Sciences Mathmatiques C.N.R.S. France -Laboratorio Ypatia delle Scienze Matematiche I.N.D.A.M. Italy (LYSM).
Date : September 19, 2020.2020
Mathematics Subject Classification.
Key words and phrases.
Noncommutative Dirichlet form, Noncommutative Potential Theory.
8. KMS-symmetric semigroups on C ∗ -algebras 388.1. KMS-states on C ∗ -algebras 389. Application to Quantum Statistical Mechanics 409.1. Heisenberg Quantum Spin Systems. 409.2. Markovian approach to equilibrium 4110. Applications to Quantum Probability 4210.1. Compact Quantum Groups d’apres S.L. Woronowicz 4210.2. L´evy processes on Compact Quantum Groups 4411. References 481. Introduction.
Our intent here is to trace some of the main steps of Noncommutative Potential theory,starting from the initial “building blocks” by Sergio Albeverio and Raphael Hoegh-Khron[AHK1,2]. The point of view adopted in treating Potential Theory it is essentially the oneof Dirichlet forms, i.e. the point of view of Energy. The justification for this is that, notonly the motivating situations to develop a potential theory on operator algebras came fromMathematical Physics but also that the concept of Energy seems to have a unifying characterwith respect to the different aspects of the subject.The present exposition is thought to be addressed to researcher not necessarily familiar withthe tools of operator algebras and, in this respect, we privileged the illustration of examplesand applications instead to provide the details of the proofs.In this presentation several aspects of the theory has been necessarily sacrificed and for themwe refer to other presentations [Cip4], [Cip5]. In particular, the construction of Fredholmmodules and Dirac operators from Dirichlet forms and the realization of Dirichlet spaces asistances of A. Connes’ Noncommutative Geometry [Co5] can be found in [Cip4],[CS3] and[S4,7] while the study of energy states, potentials and multipliers of noncommutative Dirichletspaces has been initiated in [CS4]. The details of the theory on KMS symmetric Markoviansemigroups on C ∗ -algebras can be found in [Cip5].The recent developments of the theory of noncommutative Dirichlet forms show a not raresituation in Mathematics in which a theory born to solve specific problems, as time goes by,applies to, apparently far away, others. In this respect we review in Section 7 the recent closerelationships among spectral characteristics of noncommutative Dirichlet forms and approx-imation properties of von Neumann algebras such as Haagerup Property (H), amenabilityand Property (T). In particular a new characterization of the Murray-von Neumann Prop-erty (Γ) is proved in terms of absence of a Poincar´e inequalitiy for elementary Dirichlet forms.2. Commutative Potential theory
Classical Potential Theory.
Classical Potential Theory concerns properties of theDirichlet integral D : L ( R d , m ) → [0 , + ∞ ] D [ u ] := Z R d |∇ u | dm as a lower semicontinuous quadratic form on the Hilbert space L ( R d , m ), which is finiteon the Sobolev space H ( R d ). The associated positive, self-adjoint operator is the Laplace operator ∆ = − d X k =1 ∂ k D [ u ] = k√ ∆ u k which generates the heat semigroup e − t ∆ whose Gaussian kernel e − t ∆ ( x, y ) = (4 πt ) − d/ e − | x − y | t is the fundamental solution of the heat equation ∂ t u + ∆ u = 0. The Brownian motion(Ω , P x , B t ) is the stochastic processes associated to the semigroup by the relation( e − t ∆ u )( x ) = E x ( u ◦ B t )which is also directly connected to the Dirichlet integral by the identity D [ u ] = lim t → + E m ( | u ◦ B t − u ◦ B | )2 t . The polar sets, i.e. those sets which are avoided by the Brownian motion, can be characterizedas those which have vanishing electrostatic capacity, defined in terms of the Dirichlet integralitself as Cap( A ) := inf {D [ u ] + k u k : u ∈ H ( R n ) , A ≤ u } for any open set A ⊆ R n and then asCap( B ) := inf { Cap( A ) : B ⊆ A, A open } for any other measurable set B ⊆ R n . The heat semigroup is Markovian on L ( R n , m ) in thesense that it is strongly continuous, contractive, positivity preserving and satisfies e − t ∆ u ≤ u is a real function such that u ≤
1. By these properties it can be extended to acontractive and positivity preserving semigroup on any L p ( R n , m ) for p ∈ [1 , + ∞ ] which isstrongly continuous for p ∈ [1 , + ∞ ) and weakly ∗ -continuous for p = + ∞ . The Markovianityof the heat semigroup is equivalent to the following property, also called Markovianity, of theDirichlet integral D [ u ∧ ≤ D [ u ] u = ¯ u ∈ L ( R n , m )which can be easily checked using differential calculus and the definition of the Dirichletintegral. All others above properties can be proved by the explicit knowledge of the Greenkernel of heat semigroup which, for d ≥ − u ( x ) = Z R d G ( x, y ) u ( y ) m ( dy ) G ( x, y ) = c d · | x − y | − d . Beurling-Deny Potential Theory. ([BD2], [Den], [CF], [FOT], [LJ], [Sil1], [Sil2]). Aturning point in the development of potential theory was represented by two seminal papersby A. Beurling and J. Deny [BD1,2]. They developed a kernel free potential theory based onthe notion of energy on general locally compact measured spaces (
X, m ). The whole theoryrelies on the notion of regular Dirichlet form which is required to be a lower semicontinuousquadratic functional on L ( X, m ) satisfying • Markovianity E : L ( X, m ) → [0 , + ∞ ] E [ u ∧ ≤ E [ u ] • regularity : F ∩ C ( X ) is a form core uniformly dense in C ( X ) FABIO CIPRIANI and where the form domain F := { u ∈ L ( X, m ) : E [ u ] < + ∞} is assumed to be L -dense.The lower semicontinuity of E on L ( X, m ), being equivalent to the closedness of the denselydefined quadratic form ( E , F ) on L ( X, m ), implies the existence of a nonnegative, self-adjointoperator (
L, D ( L )) which generates a Markovian semigroup e − tL on L ( X, m ).2.2.1.
Beurling Deny decomposition.
One of the first fundamental results in the Beurling-Deny analysis concerns the structure of a general regular Dirichlet form: these can be uniquelyrealized as a sum of three Markovian forms (each of which not necessarily closed) E = E d + E j + E k where the jumping part has the form E j [ u ] = Z X × X \ ∆ X | u ( x ) − u ( y ) | J ( dx, dy )for a positive measure J supported off the diagonal ∆ X of X × X , the killing part appears as E k [ u ] = Z X | u ( x ) | k ( dx )for some positive measure k on X and the diffusion part is strongly local in the sense that E d [ u + v ] = E d [ u ] + E d [ v ]whenever u is constant in a neighborhood of the support of v .Two turning point in the development of Potential Theory took place on the probabilisticside when M. Fukushima associated a Hunt stochastic process (Ω , P x , X t ) to a regular Dirichletform in such a way that E [ u ] = lim t → + E ( | u ◦ X t − u ◦ X | )2 t u ∈ F and when M. Silverstein introduced the notion of extended Dirichlet space , especially forthe connections with the boundary theory and the random time change of symmetric Huntprocesses.The process is a stochastic dynamical system which represents the semigroup through( e − tL u )( x ) = E x ( u ◦ X t ) . A basic tool in the development of the Beurling-Deny theory of a regular Dirichlet form ( E , F )is the capacity one associates to it exactly in the same way we have seen above in the caseof Dirichlet integrals. A key point to construct the associated stochastic processes is the factthat the regularity property of the Dirichlet form allows to prove that the capacity associatedto it is in fact a Choquet capacity which implies that Borel sets are capacitable.From the point of view of the process, the three different summands of the Beurling-Denydecomposition have a nice and useful probabilistic interpretation: the measure J counts thejumps of the process, the measure k specifies the rate at which the process is killed inside X and a Dirichlet form is strongly local if and only if the associated process is a diffusion, i.e.it has continuous sample paths. In Section 5.2 we will show an independent, algebraic wayto prove the above decomposition of Dirichlet forms. Operator algebras C ∗ -algebras as noncommutative topology. ([Arv], [Dix1], [Ped], [T2]).A C ∗ -algebra is A is a Banach ∗ -algebra in which norm and involution conspire as follows k a ∗ a k = k a k a ∈ A .
This notion generalizes topology in an algebraic form in the sense that, by a theorem of I.M.Gelfand, a commutative C ∗ -algebra A is isomorphic to the algebra C ( X ) of continuous func-tions vanishing at infinity on a locally compact Hausdorff space X , called the spectrum of A .In C ( X ) the product of functions is defined pointwise, the involution is given by pointwisecomplex conjugation and the norm is the uniform one.The simplest example of a noncommutative C ∗ -algebra is the full matrix algebra M n ( C ) wherethe product is the usual one rows-by-columns, the involution of a matrix is defined as its ma-trix adjoint and the norm is given by the spectral radius. Finite dimensional C ∗ -algebras are isomorphic to finite direct sums of full matrix algebras.The simplest examples of noncommutative, infinite dimensional C ∗ -algebras are those of thealgebra B ( h ) of all bounded operators and its subalgebra of all compact operators K ( h ) onan infinite dimensional Hilbert space h , the norm being the operator one.A morphism α : A → B between C ∗ -algebras A, B is a norm continuous ∗ -algebras morphism.A first example of the deep interplay that the algebraic and the analytic structures on a C ∗ -algebra give rise, is the fact that ∗ -algebra morphisms are automatically norm continuous.Morphisms between commutative C ∗ -algebras C ( X ) and C ( Y ) correspond to homeomor-phisms φ : Y → X by α ( f ) = f ◦ φ .A morphism of type π : A → B ( h ) is called a representation of A on the Hilbert space h .It is called faithful if it is an injective map and in this case A can be identified with theC ∗ -subalgebra π ( A ) ⊆ B ( h ). Any C ∗ -algebra admits a faithful representation.A C ∗ -algebras is, in particular, an ordered vector space where the closed cone is given by A + := { a ∗ a ∈ A : a ∈ A } . When A is represented as a subalgebra of some B ( h ), the positive elements of A are positive,self-adjoint operators on h . In C ( X ), the positive elements are just the nonnegative functions.3.2. von Neumann Algebras as noncommutative measure theory. ([Dix2], [MvN],[Ped]). A von Neumann algebra M is a C ∗ -algebra which admits a predual M ∗ as a Banachspace in the sense that ( M ∗ ) ∗ = M .Any commutative, σ -finite von Neumann algebra is isomorphic to the algebra of (classes of)essentially bounded measurable functions L ∞ ( X, m ) on a measured standard space (
X, m )with L ( X, m ) as predual space. This commutative situation forces to regard the theory ofvon Neumann algebras as a noncommutative generalization of Lebesgue measure theory. Evenif this is a fruitful point of view, other natural constructions suggest to look at the theory asa generalization of Euclidean Geometry and as a generalization of Harmonic Analysis.The simplest example of a noncommutative von Neumann algebra is that of the space B ( h )of all bounded operators acting on a Hilbert space h having dimension greater than one. Thepredual of B ( h ) is given by the Banach space L ( h ) of trace-class operators on h and theduality is given by h A, B i := Tr( AB ) A ∈ B ( h ) , B ∈ L ( h ) . A von Neumann algebra is σ -finite if all collections of mutually disjoint orthogonal projections have atmost a countable cardinality. von Neumann algebras acting on separable Hilbert spaces are σ -finite (theconverse being in general not true). FABIO CIPRIANI
All C ∗ -algebras are isomorphic to norm-closed subalgebras of some B ( h ) and all von Neumannalgebras are isomorphic to subalgebras of some B ( h ), closed in its weak ∗ -topology. A firstfundamental results of J. von Neumann asserts that for any subset S ⊆ B ( h ), its commutant S ′ := { a ∈ B ( h ) : ab = ab, for all b ∈ B ( h ) } is a von Neumann algebra. A second fundamental result of J. von Neumann asserts that aninvolutive subalgebra M ⊆ B ( h ) is a von Neumann algebras iff it is weakly ∗ -closed and iff itcoincides with its double commutant M = M ′′ := ( M ′ ) ′ . A key aspect is that for an involutivesubalgebra M ⊆ B ( h ) its weak ∗ -closure coincides with its double commutant ( M ′ ) ′ .The center of a von Neumann algebra is defined asCenter( M ) := { x ∈ M : xy = yx, y ∈ M } . and M is called a factor if its center reduces to the one dimensional algebra C · M of scalarmultiples of the unit of M .3.3. Weights, traces, states and the GNS representation. ([Dix1], [Ped]). A positivefunctional on a C ∗ -algebra A is a linear map τ : A → C such that τ ( a ) ≥ a ∈ A + . These are automatically bounded and are called states when having norm one. In case A hasa unit, a positive functional is a state as soon as τ (1 A ) = 1 as it follows from 0 ≤ a ≤ k a k A · A .Positive functionals are noncommutative analog of finite, positive Borel measures on locallycompact spaces: in fact, by the Riesz Representation Theorem, a positive functional on acommutative C ∗ -algebra C ( X ) corresponds, via Lebesgue integration, to a finite, positiveBorel measure m on X τ ( a ) = Z X a dm a ∈ C ( X ) , which is a probability if and only if τ is a state. To accommodate the analog of possiblyunbounded positive Borel measures, one has to consider weights on A defined as functions τ : A + → [0 , + ∞ ] which are homogeneous and additive in the sense τ ( λa ) = λτ ( a ) , τ ( a + b ) = τ ( a ) + τ ( b ) a, b ∈ A + , λ ≥ . If a weight is everywhere finite, then it can be extended to a positive linear functional on A .A weight is called a trace if it is invariant under inner automorphisms in the sense τ ( uau ∗ ) = τ ( a ) , a ∈ A + for all unitaries u ∈ e A = A ⊕ C (recall that A is a two-sided ideal in e A ). This is equivalentto require that τ is central in sense that τ ( a ∗ a ) = τ ( aa ∗ ) a ∈ A. If τ is finite this reduces to τ ( ab ) = τ ( ba ) a, b ∈ A. A weight is faithful if it vanishes τ ( a ) = 0 on a ∈ A + only when a = 0. In the commutativecase, faithful weights correspond to fully supported positive Borel measures.A weight is densely defined if the ideal A τ := { a ∈ A + : τ ( a ∗ a ) < + ∞} is dense in A . If atrace is lower-semicontinuous, than it is semifinite in the sense that τ ( a ) = sup { τ ( b ) ∈ A + : b ≤ a } b ∈ A + . On a von Neumann algebra, a weight is normal if τ (sup i ∈ I a i ) = sup i ∈ I τ ( a i )for any net { a i : i ∈ I } ⊂ A + admitting a least upper bound in A + . The predual Banachspace M ∗ of a von Neumann algebra M can be shown to the space of all normal continuousfunctionals on M . A von Neumann algebra is said to be finite (resp. semi-finite ) if, for everynon-zero a ∈ A + , there exists a finite (resp. semi-finite) normal trace τ such that τ ( a ) > properly infinite (resp. purely infinite ) if the only finite (resp. semi-finite)normal trace on A is zero. On a semi-finite von Neumann algebra, there exists a semi-finitefaithful normal trace.In this exposition we will be essentially concerned with semi-finite von Neumann algebrasand in particular with those which are σ -finite in the sense that they admit a faithful, normalstate. If h is a separable Hilbert space, then any von Neumann algebra A ⊆ B ( h ) is σ -finite.In fact, for any Hilbert base { e k ∈ h : k ∈ N } , a faithful, normal state is provided by τ ( x ) := X k ∈ N ( e k | xe k ) h x ∈ A. In a way similar to the one by which a probability measure m on X give rise to the Hilbertspace L ( X, m ) and to the representation of continuous functions in C ( X ) as multiplicationoperators on it, a densely defined weight τ on a C ∗ -algebra A give rise to a Hilbert space L ( A, τ ) on which the elements a ∈ A act as bounded operators. This is called the Gelfand-Neimark-Segal or GNS-representation of A associated to τ .In fact, the sesquilinear form x, y τ ( x ∗ y ) on the vector space A , satisfies the Cauchy-Schwarz inequality | τ ( x ∗ y ) | ≤ τ ( x ∗ x ) τ ( y ∗ y ) x, y ∈ A τ and a Hilbert space L ( A, φ ) can be constructed from the inner product space A τ by sepa-ration and completion. Since A τ is an ideal of A , the left regular action b ab of A ontoitself give rise to an action of A onto A τ and then to a representation of A on the GNSHilbert space. If τ is faithful, the identity map of A give rise to an injective, bounded map A → L ( A, τ ) and if A is unital, the vector ξ φ ∈ L ( A, τ ) image of the identity 1 A ∈ A ,allows to represent the state φ by τ ( x ) = ( ξ τ | xξ τ ) . This vector, uniquely determined by thisproperty, is cyclic in the sense that Aξ τ = L ( A, τ ) and separating in the sense that if a ∈ A and aξ τ = 0 then a = 0.The von Neumann algebra L ∞ ( A, τ ) := ( π GNS ( A )) ′′ ⊆ B ( L ( A, τ )) obtained by w ∗ -completion,is called the von Neumann algebra generated by τ on A . The GNS-representation can thenbe extended to a normal representation of L ∞ ( A, τ ). As notations are aimed to suggest, thisis a generalization of the usual construction of Lebesgue-Riesz measure theory.In the case of the trace functional Tr on K ( h ), the associated GNS space is L ( K ( h ) , Tr) = L ( h ) the space of Hilbert-Schmidt operators on which compact operators in K ( h ) act by leftcomposition.The Hilbert space of the GNS representation of a faithful trace is naturally endowed witha closed convex cone L ( A, τ ), which provides an order structure on L ( A, τ ). It is definedas the closure of A τ . In the commutative case A = C ( X ), this is just the cone of squareintegrable, positive functions. L ( X, m ). The construction of a suitable closed, convex conefrom a faithful state on a C ∗ -algebra or from a faithful normal state on a von Neumannalgebra will be done later on.We conclude this section mentioning that a noncommutative integration theory for traces on FABIO CIPRIANI C ∗ -algebras has been developed in [Ne], [Se] giving rise to an interpolation scale of spaces L p ( A, τ ) between the von Neumann algebra L ∞ ( A, τ ) and its predual L ( A, τ ). The elementsof this spaces can realized as closed operators on L ( A, τ ).3.4.
Morphisms of operator algebras. ([Ped]).The most obvious notion of morphism toform a category of C ∗ -algebras is certainly that of continuous morphisms of involutive al-gebras. However, this category risks to have a poor amount of morphisms. For example, if α : A → B is a morphism and B is commutative then α ( ab − ba ) = α ( a ) α ( b ) − α ( b ) α ( a ) = 0 sothat if the algebra generated by commutators [ a, b ] := ab − ba is dense in A then α = 0. Thisis the case for example of A = K ( h ) or more generally for the so called stable C ∗ -algebras.We illustrate now a much more well behaved notion of morphism between C ∗ and von Neu-mann algebras, i.e. completely positive map (an even more general and fundamental notionof morphism is that of Connes correspondence , which we will meet later on in this lectures).This notion is of probabilistic nature in the sense that, among commutative von Neumannalgebras, completely positive maps are just the transformations associated to positive ker-nels . Notice, en passant , that another basic tool in operator algebra theory which is of clearlyprobabilistic nature is the notion of conditional expectation . See discussion in [Co5 Chapter5 Appendix B].Beside to any C ∗ -algebra A we may consider its matrix ampliations A ⊗ M n ( C ), n ≥
1. Alinear map T : A → B is said to be completely positive , or CP map, if its ampliations T ⊗ I n : A ⊗ M n ( C ) → B ⊗ M n ( C )are positive for any n ≥ ∗ -algebra morphisms (such as representations) are completelypositive maps. If A or B is commutative, all positive maps (in particular, states) are au-tomatically completely positive. Complete positivity is however a much more demandingproperty than just positivity. While the general structure of positive maps is rather elusive,even in a finite dimensional setting, the structure of CP maps is completely described by theStinespring Theorem [Sti]. We may consider, without loss of generality, the case of a CPmap T : A → B ( h ). The result ensures the existence of a representation π : A → B ( k ) on aHilbert space k and that of a bounded operator V : h → k such that T a = V ∗ π ( a ) V a ∈ A. In case A is unital and T A = 1 A , then V ∗ V = I h so that V is an isometry which can beconsidered as an immersion of h into k . V ∗ is then the projection of k onto h and the CPmap T results as the compression of the restriction of a representation . The Steinespringconstruction can be considered as a generalization of the GNS representation. One startsendowing the vector space A ⊗ alg h by the sesquilinear form( a ⊗ ξ | b ⊗ η ) := ( ξ | T ( a ∗ b ) η ) h a, b ∈ A , ξ, η ∈ h and checks that the CP property just ensures that this form is positive definite. Cuttingout its kernel and completing the normed space obtained, one gets the Hilbert space k . Therepresentation of A on k is an ampliation of the left regular representation of A as it is inducedby the map a ( b ⊗ ξ ) ab ⊗ ξ .A positivity preserving map φ : M → N between von Neumann algebras, is normal if φ (sup α x α ) = sup α φ ( x α ) for all bounded monotone increasing nets of self adjoint elements { x α } ⊂ M . The property is equivalent to the continuity with respect to weak ∗ -topology ofthe algebras. Positivity preserving and Markovian semigroups on operator algebras. ([Br1]).A strongly continuous semigroup { T t : t > } of contractions on a unital C ∗ -algebra AT t : A → A T t ◦ T s = T t + s , T = I , lim t → + k a − T t a k A = 0 , a ∈ A is said to be Markovian if it is positivity preserving and subunital ≤ a ≤ A ⇒ ≤ T t a ≤ A a ∈ A. If A is endowed with a densely defined trace τ , the semigroup is said to be τ -symmetric if τ ( a ∗ ( T t b )) = τ (( T t a ∗ ) b ) a, b ∈ A ∩ L ( A, τ ) . In case A is a von Neumann algebra, one requires the trace to be normal and the semigroupto be point-weak*-continuous in the senselim t → + η ( a − T t a ) = 0 a ∈ A, η ∈ A ∗ . In case the C ∗ -algebra A does not have a unit, one can understand positivity preserving andMarkovianity embedding A into a larger unital C ∗ -algebra e A and there using the unit 1 e A instead of 1 A . For example one can choose A ⊕ C .The generator ( L, D ( L )) of a Markovian semigroup on a C ∗ -algebra (resp. a von Neumannalgebra) A is a norm (resp. weak*) closed, densely defined operator on A defined as D ( L ) := { a ∈ A : ∃ lim t → + a − T t at ∈ A } La := lim t → + a − T t at a ∈ D ( L )where the limit is understood in the norm (resp. weak ∗ )-topology. Norm continuous semi-groups are exactly those which have bounded generators and these are classified in [Lin], [CE]. Completely positive, completely contractive or completely Markovian semigroups are definedas those semigroups on A whose ampliations to the algebras A ⊗ M n ( C ) are positive, contrac-tive or Markovian for all n ≥
1. Completely Markovian semigroups are also called dynamicalsemigroups especially in Mathematical Physics and Quantum Probability (see [D1]).
Remark . Notice that, on von Neumann algebras, strongly continuous semigroups areautomatically norm continuous as it follows by a direct application of [E Theorem 1]. Sincesemigroups with bounded generators have rather limited applications, this is the reasons forwhich on von Neumann algebra the natural continuity of a semigroups is the point-weak*-continuity . 4.
Noncommutative Potential Theory
In this section, we let (
A, τ ) be a C ∗ -algebra endowed with a densely defined, lower semicon-tinuous faithful trace and consider the GNS representation π GNS acting on the space L ( A, τ ).We will indicate by L ∞ ( A, τ ) the von Neumann algebra ( π GNS ( A )) ′′ ⊆ B ( L ( A, τ )) generatedby A through the GNS representation.Recall that the little Lipschitz algebra is defined asLip ( R ) := { f : R → R : f (0) = 0 , | f ( t ) − f ( s ) | ≤ | t − s | , t, s ∈ R } . If a = a ∗ ∈ A and f ∈ Lip ( R ), then f ( a ) ∈ A acquires a meaning thank to the factthat C ∗ -algebras are closed under continuous functional calculus. Since, by assumption, A τ := A ∩ L ( A, τ ) is dense in A and a fortiori in L ( A, τ ), if a = a ∗ ∈ L ( A, τ ) then f ( a ) ∈ L ( A, τ ) may bedefined as the limit in L ( A, τ ) of the sequence f ( a n ) ∈ L ( A, τ )associated to a sequence a n ∈ A ∩ L ( A, τ ) converging to a in L ( A, τ ). Dirichlet forms on C ∗ -algebras with trace d’apres Albeverio-Hoegh-Krohn. Inthis section we define Dirichlet forms and Markovian semigroups on the space L ( A, τ ) anddiscuss the connection between them and the Markovian semigroups on the von Neumannalgebra L ∞ ( A, τ ), where (
A, τ ) is a C ∗ -algebra A endowed with a densely defined, lowersemicontinuous faithful trace, introduced in [AHK1]. Even if we will not discuss them in thisnotes, we mention that D. Guido, T. Isola and S. Scarlatti in [GIS] provided the extension ofthis theory to the case of non-symmetric Dirichlet forms . Definition 4.1. A Dirichlet form is a lower semicontinuous functional E : L ( A, τ ) → ( −∞ , + ∞ ]with domain F := { a ∈ L ( A, τ ) : E [ a ] < + ∞} satisfying the propertiesi) F is dense in L ( A, τ )ii) E [ a ∗ ] = E [ a ] for all a ∈ L ( A, τ ) (reality)iii) E [ f ( a )] ≤ E [ a ] for all a = a ∗ ∈ L ( A, τ ) and all f ∈ Lip ( R ) (Markovianity).A Dirichlet form is said to beiv) regular if its domain F is dense in A v) complete Dirichlet form if the ampliation E n on the algebra ( A ⊗ M n ( C ) , τ ⊗ tr n ) defined E n : L ( A ⊗ M n ( C ) , τ ⊗ tr n ) → ( −∞ , + ∞ ] E n [[ a i,j ] ni,j =1 ] := n X i,j =1 E [ a i,j ]is a Dirichlet forms for all n ≥ { T t : t > } on L ( A, τ ) is saidvi) positivity preserving if T t a ∈ L ( A, τ ) for all a ∈ L ( A, τ )vii)
Markovian if it is positivity preserving and for a = a ∗ ∈ A ∩ L ( A, τ )0 ≤ a ≤ e A = ⇒ ≤ T t a ≤ e A t > completely Markovian if the extensions T nt := T t ⊗ I n to L ( A ⊗ M n ( C ) , τ ⊗ tr n ) areMarkovian semigroups for all n ≥ Remark .
1) If in the Markovianity condition one considers as f the zero function inLip ( R ), one verifies that Dirichlet forms are nonnegative.2) It may be checked that Markovianity is equivalent to the single contraction property E [ a ∧ ≤ E [ a ] a = a ∗ ∈ L ( A, τ )in which only the unit contraction f ( t ) := t ∧ f ( a ) for a fixed a = a ∗ ∈ L ( A, τ ) and f ∈ Lip ( R ) has been shown in [AHK1] as those hermitian b = b ∗ ∈ L ( A, τ ) such that b ≤ a , | b ⊗ − ⊗ b | ≤ | a ⊗ − ⊗ a | .
4) Since L ( A ⊗ M n ( C ) , τ ⊗ tr n ) = L ( A, τ ) ⊗ L ( M n ( C ) , tr n ), the ampliations are equivalentlydefined as E [ a ⊗ m ] := E [ a ] · k m k a ⊗ m ∈ L ( A, τ ) ⊗ L ( M n ( C ) , tr n ) . The first fundamental result of the Albeverio-Hoegh-Krohn work [AHK1] is the followingcorrespondence which generalize that of Beurling-Deny in the commutative case.
Theorem 4.3.
There exists a one-to-one correspondence amongi) Dirichlet forms ( E , F ) on L ( A, τ ) ii) Markovian semigroups { T t : t > } on L ( A, τ ) iii) τ -symmetric, Markovian semigroups { S t : t > } on the von Neumann algebra L ∞ ( A, τ ) .Moreover, the semigroups are completely Markovian if and only if the quadratic form is acompletely Dirichlet form. The correspondence between semigroups and quadratic forms on L ( A, τ ) is given by therelation E [ a ] = lim t → t − (cid:0) a | ( I − T t ) a (cid:1) L ( A,τ ) a ∈ L ( A, τ )where both sides are finite precisely when a ∈ F . The correspondence between the semigroupon the Hilbert space L ( A, τ ) and the one the von Neumann algebra L ∞ ( A, τ ) is given by S t a = T t a a ∈ A ∩ L ( A, τ ) . Remark . The above correspondence is exactly the original one proved in [AHK1] even ifthe result still holds true if one start with a semi-finite von Neumann algebra (
M, τ ) and adensely defined, semifinite trace on it.We prefer the first presentation since it prepares the ground i) to naturally introduce anddiscuss the notion of regularity of a noncommutative Dirichlet forms, which, as in the Beuling-Deny theory, is the key notion to develop a rich potential theory [CS4] and ii) to develop theintrinsic differential calculus of Dirichlet spaces (see Section 5 below and [CS1]).The second fundamental result of the Albeverio-Hoegh-Krohn work is the following
Theorem 4.5.
Let the C ∗ -algebra A be represented as acting on a Hilbert space h . Let K be a self-adjoint (non necessarily bounded) operator on h and m i ∈ L ( h ) be Hilbert-Schmidtoperators for i = 1 , , · · · . Then the quadratic form E [ a ] := ∞ X i =1 Tr( | [ a, m i ] | ) + Tr( K | a | ) a ∈ L ( A, τ ) is a completely Dirichlet form provided it is densely defined. This result is fundamental not only because it provides a tool to construct a large class ofexamples but also because it suggests, at least in one direction, a correspondence betweencompletely Dirichlet forms and unbounded derivations a i [ a, m ] on the C ∗ -algebra A (seeSection 5 below).The proofs in [AHK1] of both theorems are based on a careful analysis of the normal con-tractions on A .4.1.1. Dirichlet energy forms on Clifford C ∗ -algebras. Here we illustrate the first example ofa noncommutative Dirichlet form. It has been created to represents the quadratic form of aphysical Hamiltonian of an assembly of electrons and positrons. In particular, its definitionand the study of its properties has been introduced by L. Gross [G1,2] in connection with theproblem of existence and uniqueness of the ground state of physical Hamiltonians describingFermions.Let h be a complex Hilbert space and J a conjugation on it (i.e. an anti-linear, anti-unitary operator such that J = I ). Systems whose number of particles is not a priori bounded aboveare described by the Fock space F ( h ) := ∞ M n =0 h ⊗ n . Particles system obeying a Fermi-Dirac statistics are described by the Fermi-Fock subspace F − ( h ) := P − ( F ( h )) . where the orthogonal projection P − is defined by P − ( f ⊗ · · · ⊗ f n ) = ( n !) − X π ε π f π (1) ⊗ · · · ⊗ f π ( n ) where the sum is over all permutations (1 , · · · , n ) ( π (1) , · · · , π ( n )). For f ∈ h , thecreation operator a ∗ ( f ), defined as a ∗ ( f ) := √ n + 1 P − ( f ⊗ g ⊗ · · · ⊗ g n ), is bounded withnorm k a ∗ ( f ) k = k f k h . Together with the annihilation operator defined by a ( f ) := ( a ∗ ( f )) ∗ ,it satisfies the canonical anti-commutation CAR relations a ( f ) a ( g ) + a ( g ) a ( f ) = 0 , a ( f ) a ∗ ( g ) + a ∗ ( g ) a ( f ) = ( f | g )I h f, g ∈ h which represents the Pauli’s Exclusion Principle. The Clifford C ∗ -algebra Cl ( h ) is defined asthe C ∗ -algebra generated by the fields operators b ( f ) := a ∗ ( f ) + a ( J f ) f ∈ h i.e. as the intersection of all C ∗ -subalgebras of B ( F − ( h )) containing the fields { b ( f ) : f ∈ h } .It is highly noncommutative since it is a simple C ∗ -algebra in the sense that it has no nontrivialclosed, bilateral ideals. The Fock vacuum vector Ω := 1 ⊕ · · · ∈ F − ( h ) defines a trace vectorstate on it by τ ( A ) := (Ω | A Ω) A ∈ Cl ( h )and the natural map D : Cl ( h ) → F − ( h ) given by A A Ω, extends to a unitary map from L ( Cl ( h ) , τ ) onto F − ( h ), called the Segal isomorphism. This natural isomorphism allows totransfer on the Fermi-Fock space the order structure one has on L ( Cl ( h ) , τ ) and viceversa,to study on L ( Cl ( h ) , τ ) operators originally created on F − ( h ). This procedure is especiallyuseful in combination with second quantization, where a self-adjoint operator ( A, D ( A )) on h give rise to a self-adjoint operator ( d Γ( A ) , D ( d Γ( A ))) on F − ( h ) as follows. First defineself-adjoint operators ( A n , D ( A n )) on P − ( h ⊗ n ) for n ≥ A = 0 and A n ( P − ( f ⊗ · · · ⊗ f n )) := n X k =1 P − ( f ⊗ · · · ⊗ Af k ⊗ · · · f n ) f ⊗ · · · ⊗ f n ∈ D ( A n ) := D ( A ) ⊗ n . The direct sum of the A n is essentially self-adjoint because it is symmetric and it has a denseset of analytic vectors formed by finite sums of anti-symmetrized products of analytic vectorsof A . The self-adjoint closure d Γ( A ) := ⊕ ∞ k =0 A n of this sum is called the second quantization of A and is denoted by ( d Γ( A ) , D ( d Γ( A ))). The main example of this procedure concerns theNumber operator d Γ( I ). Theorem 4.6. (Clifford Dirichlet form) Let ( A, D ( A )) be a self-adjoint operator on h , com-muting with J and satisfying A ≥ mI h for some m > . Theni) the quadratic form ( E , F ) of the operator H := D − d Γ( A ) D E [ ξ ] := ( ξ | Hξ ) L ( Cl ( h ) ,τ ) = ( Dξ | d Γ( A ) Dξ ) F − ( h ) ξ ∈ D ( H ) is a completely Dirichlet form on L ( Cl ( h ) , τ ) ;ii) the completely Markovian semigroup e − tH is hypercontractive in the sense that it is bounded from L ( Cl ( h ) , τ ) to L ( Cl ( h ) , τ ) as soon as mt ≥ (ln 3) / ,iii) inf σ ( H ) is an isolated eigenvalue of finite multiplicity. The main point is to prove the result for A = I h so that the Hamiltonian is H = Dd Γ( I h ) D − = DN D − is unitarely equivalent to the Number operator N = d Γ( I h ). Weshall see later a proof based on the structure of the Dirichlet form of the Number operator.The interest in noncommutative Dirichlet forms originated in QFT to extend to Fermions thenon perturbative techniques of E. Nelson, I.M. Segal, J. Glimm, L. Gross, A. Jaffe, B. Simonand others, elaborated for Bosons systems.We conclude this section noticing that the Clifford von Neumann algebra L ∞ ( Cl ( h ) , τ ) gener-ated by the GNS representation of the Clifford C ∗ -algebra provided by the Fock vacuum state,is isomorphic to the the hyperfinite II -factor (usually denoted by R ) to which τ extends toa normal tracial state. While the hyperfinitness (see Section 7.3 below) is a reflection of thefact that L ∞ ( Cl ( h ) , τ ) is generated by the net of finite-dimensional subalgebras correspond-ing to finite dimensional subspaces of the Hilbert space h , its uniqueness is a fundamentalresult of A. Connes [Co 3]. As any von Neumann algebra, R is generated by its projections p ∈ R (which are defined as the self-adjoint elemets p = p ∗ satisfying p = p ). However,while projections in a type I von Neumann algebra as B ( h ) have traces which can assumeintegers values only (equal to the dimension of their ranges), the trace of a projection in R may assume any real value τ ( p ) ∈ [0 , real dimension of the range of p .This is the reason by which J. von Neumann regarded R as exhibiting a Euclidean continuousgeometry .5.
Dirichlet forms and Differential Calculus: bimodules and derivations
In this section we show that on C ∗ -algebras endowed with a densely defined, lower semicon-tinuous, faithful trace ( A, τ ), completely Dirichlet forms are representations of a differentialcalculus (see [CS1], [S1,2]). In fact they can be constructed, on one side, and determine, onthe other side, closable derivations on the C ∗ -algebra A . This was suggested, at least in onedirection, by the result of Albeverio-Hoegh-Krohn illustrated in Theorem 4.5 above.To specify what a derivation on a C ∗ -algebra A is, let us recall the notion of A -bimodule H :this is an Hilbert space together with two continuous commuting actions (say left and right)of A A × H ∋ ( a, ξ ) aξ ∈ H , H × A ∋ ( ξ, b ) ξb ∈ H . The commutativity says that ( aξ ) b = a ( ξb ) for all a, b ∈ A and ξ ∈ H . If the left and rightactions coincides, aξ = ξa , then H is called a A -mono-module. Equivalently, a A -bimoduleis a representation on the Hilbert space H of the maximal or projective tensor product C ∗ -algebra A ⊗ max A ◦ . Here A ◦ denote the opposite C ∗ -algebra coinciding with A as linear spacewith involution but in which the product is reversed in order: x ◦ y ◦ := ( yx ) ◦ .A symmetric A -bimodule ( H , J ) is a A -bimodule J together with a conjugation J such that J ( aξb ) = b ∗ ( J ξ ) a ∗ a, b ∈ A, ξ ∈ H . Definition 5.1. (Derivation on C ∗ -algebras) A derivation on a C ∗ -algebra A is a linear map ∂ : D ( ∂ ) → H defined on a subalgebra D ( ∂ ) ⊆ A with values into a A -bimodule H satisfyingthe Leibnitz rule ∂ ( ab ) = ( ∂a ) b + a ( ∂b ) a, b ∈ D ( ∂ ) ⊆ A. The derivation is called symmetric if D ( ∂ ) is involutive, H is symmetric and J ( ∂a ) = ∂a ∗ a ∈ D ( ∂ ) . Here we review some examples of derivations.5.0.1.
Gradient operator.
Let M be a Riemannian manifold and consider the Hilbert space L C ( T M ) := L ( T M ) ⊗ R C obtained complexifying the Hilbert space of square integrablevector fields. This is a mono-module over the commutative C ∗ -algebra of continuous function C ( M ) where the action is defined pointwise and it can be endowed with the involution J ( χ ⊗ z ) := χ ⊗ ¯ z . If H ( M ) denotes the first Sobolev space, then a symmetric derivation isdefined by the gradient operator ∇ : C ( M ) ∩ H ( M ) → L C ( T M ) . Difference operator.
Let X be a locally compact Hausdorff space and let j be a Radonmeasure on X × X supported off the diagonal. Left and right commuting actions of C ( X )on L ( X × X, j ) may be defined as( af )( x, y ) := a ( x ) f ( x, y ) , ( f b )( x, y ) = f ( x, y ) b ( y ) a, b ∈ C ( X ) , f ∈ L ( X × X, j )and one may check that j : C c ( X ) → L ( X × X, j ) ( ∂ j a )( x, y ) := a ( x ) − a ( y )is a symmetric derivation on C ( X ) once the conjugation is defined as( J f )( x, y ) := f ( y, x ) . Killing measure.
Let X be a locally compact Hausdorff space and let k be a Radonmeasure on X . Consider L ( X, k ) as a C ( X )-bimodule where the left action is the usualpointwise one while the right action is the trivial one so that ξb := 0 for all ξ ∈ L ( X, k )and b ∈ C ( X ). If one considers as J just the pointwise complex conjugation of functions in L ( X, k ), then one may easily check that the map ∂ k : C c ( X ) → L ( X, k ) ∂a := a is a symmetric derivation on C ( X ).5.0.4. Commutators I.
Let (
A, τ ) be a C ∗ -algebra endowed with a faithful, semifinite traceand recall that A τ := { a ∈ A : τ ( a ∗ a ) < + ∞} is a bilateral ideal in A . Then if one consideron the Hilbert space L ( A, τ ) the natural left and right actions of A and the conjugation J a := a ∗ , one obtains a symmetric A -bimodule. Moreover any b ∈ A give rise to a symmetricderivation ∂ b : A τ → L ( A, τ ) ∂ b a := i [ a, b ] = i ( ab − ba ) . If a sequence { b k ∈ A : k ≥ } is fixed and one consider the direct sum of symmetric A -bimodules ⊕ ∞ k =1 L ( A, τ ), then the direct sum ∂ := ⊕ ∞ k =1 ∂ b k : D ( ∂ ) → ⊕ ∞ k =1 L ( A, τ )is a symmetric derivation defined on the involutive subalgebra D ( ∂ ) of those a ∈ A τ suchthat the series ⊕ ∞ k k [ a, b k ] k L ( A,τ ) converges. A mono-module is a bi-module in which the left and right actions coincide. Commutators II.
As a variation of the above construction, suppose that A is rep-resented on the Hilbert space h . Then the space of Hilbert-Schmidt operators L ( h ) is a A -bimodule for the left aξ and right ξb actions of a, b ∈ A on ξ ∈ L ( h ) given by compositionas operators on h . A natural involution is defined by J ξ := ξ ∗ on ξ ∈ L ( h ) and then asymmetric derivation is given by ∂ ξ : A → L ( h ) ∂ ξ a := i ( aξ − ξa ) . If a sequence { ξ k ∈ L ( h ) : k ≥ } of Hilbert-Schmidt operators is fixed and one consider thedirect sum of symmetric A -bimodules ⊕ ∞ k =1 L ( h ), then the direct sum ∂ := ⊕ ∞ k =1 ∂ ξ k : A → ⊕ ∞ k =1 L ( h )is a symmetric derivation defined on the involutive subalgebra D ( ∂ ) of those a ∈ A τ suchthat the series ⊕ ∞ k =1 k aξ k − ξ k a k L ( h ) converges. This last derivation is clearly related to theone appearing in the Albeverio-Hoegh-Khron Theorem above. We will come back on thisconstruction later on.Next result shows that closable derivations give rise to Dirichlet forms ([CS1]). Theorem 5.2.
Let ( A, τ ) be a C ∗ -algebra endowed with a densely defined, lower semicontin-uous, faithful trace and let ( ∂, D ( ∂ ) , H , J ) be a symmetric derivation, densely defined on adomain D ( ∂ ) ⊂ A ∩ L ( A, τ ) , which is closable as an operator from L ( A, τ ) to H . Then theclosure of the quadratic form E [ a ] := k ∂a k H a ∈ F := D ( ∂ ) is a completely Dirichlet form. The proof of the above result goes through the establishment of noncommutative chain rule [CS1] for closable derivation, by which one has ∂f ( a ) = (( L a ⊗ R a )( ˜ f ))( ∂a ) a = a ∗ ∈ A ∩ L ( A, τ ) , f ∈ Lip ( R )Here L a (resp. R a ) are the representation of C( sp ( a )), continuous, complex valued functionson the spectrum sp ( a ) of a , uniquely defined by L a ( f ) ξ = ( f ( a ) ξ if f (0) = 0 ξ if f ≡ f ∈ C ( sp ( a )) ξ ∈ H and R a ( f ) ξ = ( ξf ( a ) if f (0) = 0 ξ if f ≡ f ∈ C ( sp ( a )) ξ ∈ H .L a ⊗ R a is the tensor product representation of C ( sp ( a )) ⊗ C ( sp ( a )) = C ( sp ( a ) × sp ( a )). When I ⊆ R is a closed interval and f ∈ C ( I ), we will denote by ˜ f ∈ C( I × I ) the differentialquotient on I × I , sometimes called the quantum derivative of f , defined by(5.1) ˜ f ( s, t ) = ( f ( s ) − f ( t ) s − t if s = tf ′ ( s ) if s = t. Since commutators in A are bounded derivations in the above sense, the above result providesan independent proof of the Albeverio-Hoegh-Khron Theorem 4.5 above. A derivation for the Clifford-Dirichlet form.
As a further application of the aboveresult, let us show that the quadratic form E N of the Number operator N of Fermions, whenseen on the space L ( Cl ( h ) , τ ) via the Segal isomorphism D , is a completely Dirichlet formon the Clifford algebra ( Cl ( h ) , τ ). Recall first that N and E N can be written as N = ∞ X k =1 a ∗ ( f k ) a ( f k ) E N [ ψ ] := ∞ X k =1 k a ( f k ) ψ k for any orthonormal base of { f k : k ≥ } ⊂ h . Let us denote by L b and R b the left andright actions on L ( Cl ( h ) , τ ) of an element of the Clifford algebra b ∈ Cl ( h ). The symmetry β := Γ( − I h ) of F − ( h ) induces an idempotent automorphism γ ∈ B ( F − ( h )) γ ( A ) := βAβ A ∈ B ( F − ( h )) . Since b ( f ) β = − b ( f ), we have γ ( b ( f )) = − b ( f ) for all f ∈ h so that γ leaves Cl ( h ) globallyinvariant and then γ ∈ Aut( Cl ( h )). Since Cl ( h ) ⊂ L ( Cl ( h ) , τ ) and β Ω = Ω we have D ( γ ( b )) = βbβ Ω = βb Ω = βD ( b ) so that γ ( b ) = ( D − βD )( b ) b ∈ Cl ( h ) . For f ∈ h and ξ ∈ Cl ( h ) we have DL b ( f ) D − ( Dξ ) = D ( L b ( f ) ξ ) = L b ( f ) ( ξ Ω) = L b ( f ) ( Dξ ) sothat, since Cl ( h ) is dense in L ( Cl ( h ) , τ ), we obtain DL b ( f ) D − = a ∗ ( f ) + a ( J f ) = b ( f ) . Since β ( P − ( f ⊗ · · · ⊗ f n )) = P − β ( f ⊗ · · · ⊗ f n ) = ( − n P − ( f ⊗ · · · ⊗ f n ) and β Ω = Ω, by theCAR relations we have that a ∗ ( f ) − a ( J f ) β commutes with all of b ( g ) for g ∈ h and then withall elements of the Clifford algebra. Then, for b ∈ Cl ( h ) we have ( a ∗ ( f ) − a ( J f )) β ( D ( b )) =( a ∗ ( f ) − a ( J f )) β ( b Ω) = b ( a ∗ ( f ) − a ( J f )) β Ω = b ( a ∗ ( f ) − a ( J f ))Ω = ba ∗ ( f )Ω = b ( a ∗ ( f ) + a ( J f ))Ω = bb ( f )Ω = ( R b ( f ) b )Ω = D ( R b ( f ) b ) and since Cl ( h ) is dense in L ( Cl ( h ) , τ ), weobtain DR b ( f ) D − = a ∗ ( f ) − a ( J f ) β which can be rewritten as a ∗ ( f ) − a ( J f ) = DR b ( f ) D − β = DR b ( f ) ( D − βD ) D − . By summation we have a ( J f ) = 12 D ( L b ( f ) − R b ( f ) ( D − βD )) D − and changing f with J f we get a ( f ) = D (cid:16)
12 ( L b ( Jf ) − R b ( Jf ) ( D − βD ) (cid:17) D − . Consider now on the Hilbert space H γ := L ( Cl ( h ) , τ ) the Cl ( h )-bimodule structure wherethe right action of Cl ( h ) is the usual one while the left one is twisted by the automorphism γb · ξ · b := γ ( b ) ξb b , b ∈ Cl ( h ) , ξ ∈ L ( Cl ( h ) , τ )(dots indicate actions in this new bimodule structure). The definition is justified by the factthat, as one can easily check, this new left action is continuous and commutes with the rightone. Let us now check that for any fixed f ∈ h , the map ∂ f : Cl ( h ) → H γ ∂ f := i L b ( Jf ) − R b ( Jf ) ( D − βD )) , more explicitly given by a module commutator ∂ f ( b ) = 12 ( L b ( Jf ) − R b ( Jf ) ( D − βD ))( b )= 12 ( L b ( Jf ) ( b ) − R b ( Jf ) ( D − βD )( b ))= 12 ( b ( J f ) b − R b ( Jf ) ( γ ( b ))= 12 ( b ( J f ) b − γ ( b ) b ( J f ))= 12 ( b ( J f ) · b − b · b ( J f ))is a derivation in the sense that the Leibniz rule holds true: ∂ f ( ab ) = 12 { b ( J f ) · ( ab ) − ( ab ) · b ( J f ) } = 12 { ( b ( J f ) · a − a · b ( J f )) · b + a · ( b ( J f ) · b − b · b ( J f )) } = ( ∂ f a ) · b + a · ( ∂ f b ) a, b ∈ Cl ( h ) . To define a symmetry J γ such that ( H γ , J γ ) is a symmetric bimodule, notice first that theautomorphism γ of the Clifford algebra Cl ( h ) can be extended to a symmetry acting onthe Hilbert space L ( Cl ( h ) , τ ). In fact as β Ω = Ω, the trace τ is γ -invariant: τ ( γ ( b )) =(Ω | γ ( b )Ω) = ( β Ω | bβ Ω) = (Ω | b Ω) = τ ( b ) for all b ∈ Cl ( h ). Consequently γ is isometric withrespect to L -norm k γ ( b ) k = τ (( γ ( b )) ∗ γ ( b )) = τ ( γ ( b ∗ ) γ ( b )) = τ ( γ ( b ∗ b )) = τ ( b ∗ b ) = k b k b ∈ Cl ( h )so that by density it extends to an isometry on the whole L ( Cl ( h ) , τ ) such that γ = I .Further, as for all a, b ∈ Cl ( h ) we have( γ ( a ) | b ) = τ (( γ ( a )) ∗ b ) = τ (( γ ( a ∗ )) b ) = τ ( γ ( a ∗ γ ( b ))) = τ ( a ∗ γ ( a ∗ )) = ( a | γ ( b )) , it results that γ is a symmetry on L ( Cl ( h ) , τ ) which commutes with the symmetry J γ ∗ = γ , γ = I , γ ◦ J = J ◦ γ .
A new conjugation is then defined by J γ := J ◦ γ = γ ◦ J on H γ in such a way that ( H γ , J γ )is a symmetric bimodule over the Clifford algebra Cl ( h ), as it results from the followingidentities for a, b, c ∈ Cl ( h ) J γ ( a · b · c ) = J ( γ ( γ ( a ) bc )) = J ( aγ ( b ) γ ( c ))) = ( γ ( c )) ∗ ( γ ( b )) ∗ a ∗ = ( γ ( c ∗ ))( J γ ( b ) a ∗ = c ∗ ·J γ ( b ) · a ∗ . Since for f ∈ h one has J γ ( b ( J f )) = γ ( b ( J f ) ∗ ) = γ ( b ( f )) = − b ( f ), it follows that J γ ( ∂ f b ) = 12 J γ (( b ( J f ) · b − b · b ( J f )))= 12 ( b ∗ · J γ ( b ( J f )) − J γ ( b ( J f )) · b ∗ )= 12 ( b ( f ) · b ∗ − b ∗ b ( f ))= ∂ Jf ( b ∗ ) . Consequently, if f ∈ h is J -real (in the sense that J f = f ) then ∂ f is J γ -symmetric J γ ( ∂ f b ) = ∂ J ( b ∗ ) b ∈ Cl ( h ) . Choosing a Hilbert base { f k : k ≥ } ⊂ h made by J -real vectors J f k = f k , we can representthe quadratic form of the Number operator on the space L ( Cl ( h ) , τ ) as E Cl [ b ] := ∞ X k =1 k ∂ f k b k H γ b ∈ F Cl where the form domain is obviously F Cl := D − ( D ( √ N )). Setting H Cl := L ∞ k =1 H γ as adirect sum of symmetric Cl ( h )-bimodules, we have that ∂ Cl := L ∞ k =1 ∂ f k is a symmetricderivation of the Clifford algebra into H Cl such that E Cl [ b ] = k ∂ Cl b k H Cl b ∈ F Cl and which is densely defined and closable as sum of bounded derivations. By Theorem 5.2above, the associated semigroup is thus completely Markovian.5.0.7. Dirichlet form on noncommutative tori.
This is a fundamental example appearing inNoncommutative Geometry [Co5] in which the relevant algebra of coordinates A θ of the spaceis a noncommutative deformation of the algebra of continuous functions on a classical torus.For any fixed θ ∈ [0 , A θ , called noncommutative -torus , is defined as the universal C ∗ -algebra generated by two unitaries U and V , satisfying the relation V U = e iπθ U V .
When θ = 0 one recovers the algebra of continuous functions on the 2-torus. All elements of A θ can be written as a series P m,n ∈ Z c m,n U m V n with complex coefficients. A tracial state isspecified by τ : A θ → C τ ( U m V n ) = δ m, δ n, m, n ∈ Z so that L ( A θ , τ )) = n X m,n ∈ Z c m,n U m V n : X m,n ∈ Z | c m,n | < + ∞ o ≃ l ( Z ) . A densely defined, closed form is given by E h X m,n ∈ Z c m,n U m V n i = X m,n ∈ Z ( m + n ) | c m,n | on the domain F ⊂ l ( Z ) where the above series converges (i.e. the first Sobolev space). Tocheck that we are dealing with a Dirichlet form, one may observe that it is a ”square of aderivation”, taking values in the direct sum of two standard bimodules L ( A, τ ) ⊕ L ( A, τ )and given by the direct sum ∂ ( a ) = ∂ ( a ) ⊕ ∂ ( a )of the derivations ∂ and ∂ from A θ into L ( A θ , τ ) defined by ∂ ( U m V n ) = imU m V n , ∂ ( U m V n ) = inU m V n n, m ∈ Z . The associated Markovian semigroup { T t : t ≥ } , characterized by T t ( U m V n ) = e − t ( m + n ) U m V n m, n ∈ Z , is clearly conservative, in the sense that T t A θ = 1 A θ , because E [1 A θ ] = 0. Even if, at aHilbert space level, the Dirichlet form and its associated Markovian semigroup are clearlyisomorphic to the Dirichlet integral and the heat semigroup on the classical (commutative)torus T , written in Fourier series terms, their potential theoretic properties arise from theorder structure of A θ which may differ completely from those of C ( T ). The properties of A θ depend a lot upon the rationality/irrationality and diophantine approximation properties of the parameter θ ∈ [0 , S := U + U ∗ + V + V ∗ ∈ A θ . If θ = 0 we have A θ = C ( T ) so that since T is connectedand S a real continuous function, its spectrum is a closed interval of the real line. When θ isirrational, however, the spectrum of S is typically a Cantor set (so that, under ”commutativespectacles” we would look at A θ as a rather fragmented space).5.1. The derivation determined by a Dirichlet form.
The following result, in combina-tion with the previous one, establishes a one-to-one correspondence between Dirichlet formsand closable derivations on C ∗ -algebras ([CS1], [S1,2]). It says that derivations are differentialsquare roots of Dirichlet forms. It can be considered as a generalization of the constructionof the (differential first order) Dirac operator from the (differential second order) Hodge-deRham Laplacian of a Riemannian manifold. Theorem 5.3.
Let ( E , F ) be a completely Dirichlet form on a C ∗ -algebra endowed with adensely defined, lower semicontinuous faithful trace ( A, τ ) Then there exists a densely defined, L -closable derivation ( ∂, D ( ∂ )) with values in a symmetric A -bimodule ( H , J ) such that D ( ∂ ) is a form core and E [ a ] = k ∂a k H a ∈ D ( ∂ ) . The bimodule ( H , J ) is called the tangent bimodule associated to ( E , F ).5.1.1. Conditionally negative definite functions and Dirichlet forms on dual of discrete groups[CS1].
Let λ : Γ → B ( (Γ)) be the left regular representation of a countable discrete group Γ( λ s f )( t ) := f ( s − t ) s, t ∈ Γand consider the reduced C ∗ -algebra C ∗ r (Γ) defined as the smallest C ∗ -subalgebra of B ( L (Γ))containing the unitary operators λ s for all s ∈ Γ. Extending λ to c c (Γ) as a convolution( λ ( f ) g )( t ) := ( f ∗ g )( t ) = X s ∈ Γ f ( ts − ) g ( s ) f, g ∈ c c (Γ) , t ∈ Γ , we can identify c c (Γ) as a dense involutive subalgebra of C ∗ r (Γ). The involution of an element f ∈ c c (Γ) is given by f ∗ ( s ) = f ( s − ). A faithful tracial state is determined by τ ( λ ( f )) = f ( e ) f ∈ c c (Γ)where e ∈ Γ is the unit of the group. Since( λ ( f ) | λ ( g )) L ( C ∗ r (Γ) ,τ ) = ( f | g ) l (Γ) f, g ∈ c c (Γ) , we can identify L ( C ∗ r (Γ) , τ ) with l (Γ) at the Hilbert space level and represent the positivecone L ( C ∗ r (Γ) , τ ) as the one of all square integrable, positive definite functions on Γ. Thevon Neumann algebra L (Γ) generated by the unitaries { λ s : s ∈ Γ } on l (Γ) is called the group von Neumann algebra of Γ. It is a finite von Neumann algebra since the tracial state τ on C ∗ r (Γ) extends to a normal tracial state on it and it is a factor if and only if the conjugacyclass { sts − ∈ Γ : s ∈ Γ } of any t ∈ Γ is an infinite set. In this setting, any conditionally negative definite function l : Γ → C , i.e. a normalized, symmetric function l ( e ) = 0 , l ( s − ) = l ( s ) s ∈ G satisfying, for s , . . . , s n ∈ Γ, c , . . . , c n ∈ C , n X k =1 c j l ( s − j s k ) c k ≤ n X k =1 c k = 0 , determines a completely Dirichlet form E l [ a ] := X s ∈ Γ l ( s ) | a ( s ) | a ∈ l (Γ)defined on the domain F l where the above sum converges and whose associated completelyMarkovian semigroup is given by( T t a )( s ) = e − tl ( s ) a ( s ) a ∈ l (Γ) . To describe the associated derivation recall that any negative definite function can be rep-resent by a 1-cocycle c : Γ → H π of a unitary representation π : Γ → H π , i.e. a functionsatisfying c ( st ) = c ( s ) + π ( s )( c ( t )) s, t ∈ Γ , as follows l ( s ) = k c ( s ) k H π s ∈ Γ . On the Hilbert space H π ⊗ l (Γ) a C ∗ r (Γ)-bimodule structure is then specified by the leftaction π ⊗ λ and by the right action id ⊗ ρ where ρ is the right regular action of Γ and id is the trivial action on H π assigning the identity operator to any element of Γ. Using thenatural isomorphism H π ⊗ l (Γ) ≃ l (Γ , H π ), the derivation representing E l is identified by( ∂ l a )( s ) = a ( s ) c ( s ) a ∈ c c (Γ) , s ∈ Γ . When Γ = Z n , the C ∗ -algebra C ∗ r (Γ) coincides with the algebra C ( T n ) of continuous functionson the torus and if one considers the negative definite function l ( z , · · · , z n ) := | z | + · · · + | z n | ,one recovers as Dirichlet form just the standard Dirichlet integral on T n (see Section 2.1).5.2. Decomposition of derivation, Beurling-Deny fromula revisited. ([CS1,2]).Since an A -bimodule is just a representation of the C ∗ -algebra A ⊗ max A , one disposesof all the tools that representation theory offers, such has decomposition theory, to analyzederivations and their associated Dirichlet forms. In the commutative situation, for example,one obtains, by an algebraic approach, a refinement of the Beurling-Deny decomposition ofDirichlet forms.In this section we introduce notions of bounded, approximately bounded and completelyunbounded derivations and we prove that any derivation canonically split into a sum of thelatter.Let ∂ : D ( ∂ ) → H be a densely defined derivation on a C ∗ -algebra A and denote by L A ( H )the algebra of bounded operators on H which commute both with left and right actions of A .An element B ∈ L A ( H ) will be said to be ∂ -bounded if the map B ◦ ∂ extends to a boundedmap from A into H . Notice that if this is the case, B ◦ ∂ is a derivation. A projection p ∈ L A ( H ) will be said to be approximately ∂ -bounded if it is the increasing limit of a net of ∂ -bounded projections. As H is assumed to be separable, this means that one can write the A -bimodule p ( H ) as an at most countable direct sum L n H n of A -bimodules such that p ◦ ∂ decomposes as a direct sum L n ∂ n of bounded derivations ∂ n := p n ◦ ∂ where p n ∈ L A ( H )is the ∂ -bounded projection onto the A -submodule H n . A projection p ∈ L A ( H ) will besaid to be completely ∂ -unbounded if 0 is the only ∂ -bounded projection smaller than p .The derivation ∂ will be said to be bounded (resp. approximately bounded, resp. completely unbounded) if the identity operator 1 H is a ∂ -bounded (resp. approximately ∂ -bounded, resp.completely ∂ -unbounded) projection.Then one can prove that there exists a greatest approximately ∂ -bounded projection P a ∈L A ( H ) and that every ∂ -bounded B ∈ L A ( H ) satisfies B ◦ P a = B .Setting H a := P a ( H ) and H c := (1 − P a )( H ) we have the decomposition of H = H a ⊕ H c intoits approximately bounded and completely unbounded sub- A -bimodules. Correspondingly,setting ∂ a := P a ◦ ∂ and ∂ c := (1 − P a ) ◦ ∂ we have the decomposition of the derivation ∂ = ∂ a ⊕ ∂ c into its approximately bounded and completely unbounded components. Finally,any Dirichlet form can be canonically splitted as a sum of its approximately bounded andcompletely unbounded parts E [ a ] = k ∂a k H = k ∂ a a k H j + k ∂ c a k H c a ∈ F . This is a (purely algebraic) generalization of the Beurling-Deny decomposition of Dirichletforms on a commutative C ∗ -algebra A = C ( X ). In particular the completely unboundedpart E c can be identified with the diffusive part and the approximately bounded part E a correspond to the sum E j + E k of the jumping and killing parts. In the commutative situation E c can also be characterized as the part of the Dirichlet form whose C ( X )-bimodule H c is thelargest sub- C ( X )-mono-module of the C ( X )-bimodule H corresponding to E , i.e. the largestsubmodule on which the left and right actions coincide. Moreover, since as any C ( X )-mono-module, H c is the direct integral R X H x µ ( dx ) of C ( X )-mono-modules H x whose actions arethe simplest possible aξ = ξa = a ( x ) ξ a ∈ C ( X ) , ξ ∈ H x , x ∈ X, in the corresponding splitting ∂ = R X ∂ x µ ( dx ), the derivations ∂ x of C ( X ) satisfy the Leibnizproperty ∂ x ( ab ) = ( ∂ x a ) b ( x ) + a ( x )( ∂ x b ) a, b ∈ F ∩ C ( X ) . Noncommutative potential theory and curvature in Riemannian Geometry. ([CS2]). Classical Potential Theory arose to understand properties of the potential energyfunctions in electromagnetism and in classical gravity. The properties of these functions wereencoded in properties of the Laplace-Beltarmi operators and in those of the Dirichlet integralsof Euclidean domains. Dealing with Nonlinear Elasticity or Riemannian Geometry, one isnaturally lead to consider other Laplace type operators and associated quadratic energy formsacting on sections of vector bundles over Riemannian manifolds. In this section we describebriefly the strict relation between curvature and a distinguished noncommutative Dirichletform.Topological and geometric aspects of a Riemannian manifold (
V, g ) are related to theHodge-de Rham operator ∆
HdR = dd ∗ + d ∗ d acting on the space L (Λ ∗ ( V )) of square inte-grable sections of the exterior bundle Λ ∗ ( V ). It generalizes the Laplace-Beltrami operatoracting on functions but its quadratic form cannot be directly considered as a Dirichlet form,essentially because exterior forms do not realize a C ∗ -algebra. However, the geometric aspectsof V are more deeply connected to the Dirac operator D and its square D , the so calledDirac Laplacian, acting on sections of the Clifford bundle Cl ( V ) essentially because it is theconstruction of this space and operators that depends on the metric g .Recall that the fibers of Cl ( V ) are the Clifford algebras Cl ( T x V ) of the Hilbert space ( T x V, g x ).Since the exterior algebra Λ ∗ x V is nothing but the antisymmetric Fock space F − ( T x V ), global-izing the Segal isomorphism, we met in a previous section, we have a canonical isomorphismof vector bundles between Cl ( V ) and Λ ∗ ( V ). The difference is that, by construction, the fibers of the bundle Cl ( V ) form now C ∗ -algebras. As a consequence, the space C ∗ ( V, g ) ofcontinuous sections vanishing at infinity of the Clifford bundle form, by pointwise producton V , a C ∗ -algebra naturally associated to the Riemannian manifold ( V, g ). Moreover, de-noting by Ω x ∈ F − ( T x V ) the vacuum vector and by τ x ( · ) = (Ω x | · Ω x ) the associated traceon Cl ( T x V ), using the Riemannian measure m g , we get on the Clifford ∗ -algebra a denselydefined, lower semicontinus, faithful trace τ ( ω ) := Z V m g ( dx ) τ x ( ω x )and the ordered Hilbert space L ( Cl ( V, g ) , τ ). The Levi-Civita connection of ( V, g ) can belifted to a metric connection on the Clifford bundle and represented, at the analytical level,by the covariant derivative ∇ acting between the smooth sections of the Hermitian bundles Cl ( V, g ) and Cl ( V, g ) ⊗ T ∗ V . Theorem 5.4. ([DR1,2], [SU]). The closure of the Bochner integral E B [ ω ] := Z V |∇ ω ( x ) | m g ( dx ) ω ∈ C ∞ c ( Cl () V, g ) is a completely Dirichlet form on L ( Cl ( V, g ) , τ ) . The self-adjoint operator ∆ B := ∇ ∗ ∇ associated to E B , called the Bochner or connectionLaplacian, thus generates a completely Markovian semigroup on L ( Cl ( V, g ) , τ ) which isstrongly Feller in the sense that it reduces to a strongly continuous Markovian semigroup onthe Clifford algebra C ∗ ( V, g ).We may base the proof of the above result on Theorem 5.2 above. Notice first that on theHilbert space L ( Cl ( V, g ) ⊗ T ∗ V ) we may consider the C ∗ ( V, g )-bimodule structure given by σ · ( σ ⊗ ω ) · σ := ( σ · σ · σ ) ⊗ ω and a symmetry J given by J ( σ ⊗ ω ) := σ ∗ ⊗ ¯ ω for σ , σ ∈ C ∗ ( V, g ) and σ ⊗ ω ∈ L ( Cl ( V, g ) ⊗ T ∗ V ), where σ → σ ∗ is the extension of theinvolution on the Clifford algebra and ω → ¯ ω is the involution on the complexified cotangentbundle. Denoting by i X the contraction operator with respect to a smooth vector field X , wemay consider the covariant derivative along X given by ∇ X := i X ◦ ∇ . Since, by definition,the Levi-Civita connection is a metric connection, we have the identity ∇ ( f σ ) = σ ⊗ df + f ∇ σ X ( σ | σ ) = ( ∇ X σ | σ ) + ( σ |∇ X σ )for any smooth section σ of the Clifford bundle and any smooth function f . Since thecontraction i X commutes with actions of the Clifford algebra we have i X ( ∇ ( σ · σ )) = i X (( ∇ σ ) · σ + σ · ( ∇ σ ))for all smooth vector fields X so that ∇ ( σ · σ ) = ( ∇ σ ) · σ + ( ∇ σ ) · σ, from which the Leibniz property follows by polarization. Notice that this result is independentupon the Riemannian curvature of the manifold. The situation changes drastically if weconsider the Dirac Laplacian D on L ( Cl ( V, g ) , τ ) or better the Dirac quadratic form E D [ σ ] := Z V m g ( dx ) | Dσ ( x ) | . Recall that the Dirac operator D is defined locally for smooth sections σ by( Dσ )( x ) := n X k =1 e k ( x ) · ( ∇ e k σ )( x ) , where the vector fields e k form an orthonormal base in a neighborhood of x ∈ V . At anHilbert space level, the Dirac operator on the Clifford bundle is isomorphic to the de Rhamoperator on the exterior bundle D ≃ d + d ∗ and the Dirac Laplacian is isomorphic to the Hodge-de Rham Laplacian D ≃ ( d + d ∗ ) = dd ∗ + d ∗ d. Differently from the Bochner Laplacian, the potential theoretic properties of D depend,however, strongly on the sign of the curvature Theorem 5.5. ([CS2]) The following conditions are equivalenti) the Dirac quadratic form E D on L ( Cl ( V, g ) , τ ) is a completely Dirichlet formii) the curvature operator of V is nonnegative b R ≥ iii) Dirac heat semigroup e − tD on the Clifford algebra C ∗ ( V, g ) is completely Markovian. To describe the main steps of the proof let us recall that the curvature tensor R of themetric defines the curvature operator b R on the Hermitian bundle Λ V as follows b R x : Λ x V → Λ x V ( b R x ( v ∧ v ) | v ∧ v ) Λ x V := R x ( v , v , v , v ) v , v , v , v ∈ T x V. The curvatures identities imply that b R x is symmetric and thus self-adjoint when extended onthe Hilbert space obtained complexifying Λ x V . By the Bochner identity we have E D = E B + Q R where Q R is the quadratic form on L ( Cl ( V, g ) , τ ) given by Q R [ σ ] = Z V m ( dx ) Q R ( x )[ σ x ] Q R ( x )[ σ x ] = n ( n − / X α =1 µ α ( x ) k [ η α ( x ) , σ x ] k where the norms are those of the Hilbert spaces L ( Cl ( T x V, g x ) , τ x ), η α ( x ) ∈ Λ x V are eigen-vectors of b R x and µ α ( x ) ∈ R the corresponding eigenvalues. To prove that i) implies ii) onenotices that, by the Albeverio-Hoegh-Khron Theorem 4.5 or by the fact that commutatorsare bounded derivations, if the curvature operator is nonnegative, then all the eigenvaluesare nonnegative and Q R appears as a superposition of Dirichlet forms. Since, by the Davies-Rothaus Theorem 5.4 above, E B is a Dirichlet form, we have that E D is a Dirichlet form too.The proof that i) implies ii) the main idea is to use the decomposition theory of derivationsto prove that Q R is a Dirichlet forms because it coincides with the approximately boundedpart of the Dirichlet form E D . Then, a careful analysis of the structure of the Dirichlet formson the Clifford algebras of finite dimensional Euclidean spaces allows to conclude that all theeigenvalues or the curvature operator are nonnegative. Since D is, by construction, a sym-metric operator on L ( Cl ( V, g ) , τ ), if we assume that the Dirac heat semigroup is completelyMarkovian on the Clifford algebra C ∗ ( V, g ) then we get that it is a completely Markovian on L ( Cl ( V, g ) , τ ) so that the quadratic form E D is completely Dirichlet. To prove that b R ≥ ellipticity of D to deduce that e − tD transforms com-pactly supported smooth sections of the Clifford bundle into bounded continuous sections ii) Markovianity , to reduce the problems to the algebra C ( V ) of continuous functions and iii)the fact that b R ≥ C ( V ) theFeller property holds true by a classical result.5.4. Voiculescu Dirichlet form in Free Probability.
In this section we describe a Dirich-let form appearing in Free Probability Theory discovered by Dan Virgil Voiculescu [V].Let (
M, τ ) be a noncommutative probability space, i.e. a von Neumann algebra endowed witha faithful, normal trace state. Let us fix a unital ∗ -subalgebra 1 M ∈ B ⊂ M and a finite set X := { X , ..., X n } ⊂ M of noncommutative random variables, i.e. self-adjoint elements of M , algebraically free with respect to B . Let us consider the ∗ -subalgebra B [ X ] ⊂ M generatedby X and B (regarded as the algebra of noncommutative polynomials in the variables Xwith coefficients in the algebra B ) and the von Neumann subalgebra N ⊂ M generatedby B [ X ]. Let HS ( L ( N, τ )) = L ( N, τ ) ⊗ L ( N, τ ) be the Hilbert N -bimodule of Hilbert-Schmidt operators on L ( N, τ ) and 1 M ⊗ M ∈ HS ( L ( N, τ )) the rank one projection ontothe multiples of the unit 1 M ∈ M ⊂ L ( M, τ ).Within this framework, D.V. Voiculescu introduced a natural differential calculus with asso-ciated Dirichlet form.
Theorem 5.6.
There exists a unique derivation ∂ X j : B [ X ] → HS ( L ( M, τ )) for any fixed j = 1 , · · · , n such thati) ∂ X j X k = δ jk M ⊗ M k = 1 , · · · , n ii) ∂ X j b = 0 for all b ∈ B .iii) The derivation ( ∂ X j , B [ X ]) is densely defined in L ( M, τ ) , symmetric and it is closable if M ⊗ M ∈ D ( ∂ ∗ X j ) .iv) If M ⊗ M ∈ T nj =1 D ( ∂ ∗ X j ) the quadratic form ( E , F ) defined as E [ a ] := n X j =1 k ∂ X j a k HS ( L ( M,τ )) a ∈ F := B [ X ] is closable and its closure is completely Dirichlet form. Under the assumption 1 M ⊗ M ∈ T nj =1 D ( ∂ ∗ X j ), the Noncommutative Hilbert Transform of X with respect to B is defined as I ( X : B ) := (cid:16) n X j =1 ∂ X j (cid:17) M ⊗ M ∈ L ( M, τ )and the
Relative Free Fisher information of X with respect to B is defined asΦ ∗ ( X : B ) := kI ( X : B ) k L ( M,τ ) . It has been shown by P. Biane [Bia1] that this Dirichlet form is the Hessian of the RelativeFree Fischer information on the domain where the Relative Free Fisher information is finite.Moreover, if B = C and still under the assumption that 1 M ⊗ M ∈ T nj =1 D ( ∂ ∗ X j ), one has thefollowing surprising spectral characterization of semicircular random variables X Theorem 5.7. ([Bia1]) A Free Poincar´e inequality holds true for some c > c · k a − τ ( a ) k ≤ E [ a ] a ∈ L ( M, τ ) if and only if the random variable X is centered, it has unital covariance and semicirculardistribution. In the case of semi-circular systems, the self-adjoint operator associated to the Dirichletform E is unitarily equivalent to the Number operator on the Free Fock space, it generatesthe Free Ornstein-Uhlenbeck semi-group and a logarithmic Sobolev inequality holds true (see[Bia2]).6. Dirichlet forms on standard forms of von Neumann algebras
The theory of noncommutative Dirichlet forms illustrated so far has been introduced by S.Albeverio and R. Hoegh-Krohn and developed independently by J.-L. Sauvageot [S1,2,3,4,5,6,7]and by M. Lindsay and E.B. Davies [DL1,2]. We have seen that it can be applied to severalfields in which the relevant algebra of observables, to retain a physical language, is no morecommutative but it requires, however, that the reference weight or state is a trace.In this section we describe the extension of the theory to cases in which the reference func-tional is a normal state on a von Neumann algebra. In a forthcoming section we will describehow this theory may be used to study KMS-symmetric semigroups on C ∗ -algebras as it isrequired for applications to Quantum Statistical Mechanics, Quantum Field Theory, Quan-tum Probability and Noncommutative Geometry. Notice that by a fundamental result ofG.F. Dell’Antonio [Del], the von Neumann algebras appearing in Quantum Field Theory aretypically of type III so that no normal trace is available on them.The exposition is based on the approach given in [Cip1,3] which work on general standardforms of σ -finite von Neumann algebras. An approach based on the Haagerup standard form[H2] of von Neumann algebras is given in [GL1,2]. This last one has been generalized in [GL3]to consider the reference positive functional on a von Neumann algebra to be a weight. Inthis respect one ought to consult also [SV Appendix] for the correction of some of the resultsin [GL3].The Potential Theory developed by A. Beurling and J. Deny and in particular the oneto one correspondence between Dirichlet forms and symmetric Markovian semigroups on ameasured space ( X, m ), relies on the geometric properties of the cone L ( X, m ) of positivefunctions in the Hilbert space L ( X, m ). This is a closed, convex cone which is self-polar inthe sense that a ∈ L ( X, m ) if and only if ( a | b ) ≥ b ∈ L ( X, m ) . The theory of noncommutative Dirichlet forms developed by S. Albeverio and R. Hoegh-Khron on C ∗ -algebras endowed with a faithful, semifinite trace ( A, τ ) is based on analogousproperties of the cone L ( A, τ ) defined as the closure in the GNS Hilbert space L ( A, τ )of the cone { a ∈ A + : τ ( a ) < + ∞} . This cone determines, in particular, an anti-unitaryinvolution J τ on L ( A, τ ) which extends the isometric involution a a ∗ of A to the vonNeumann algebra L ∞ ( A, τ ). The whole structure ( L ∞ ( A, τ ) , L ( A, τ ) , L ( A, τ ) , J τ ) realizes the standard form of the von Neumann algebra L ∞ ( A, τ ) in the following sense.
Definition 6.1. (Standard form of a von Neumann algebra) ([Ara], [Co1], [H1]).A standard form ( M , H , P , J ) of a von Neumann algebra M acting on a Hilbert space H ,consists of a self-polar cone P ⊂ H and an anti-unitary involution J , satisfying:i) J M J = M ′ ;ii) J xJ = x ∗ , ∀ x ∈ M ∩ M ′ (the center of M );iii) J η = η , ∀ η ∈ P ;iv) xJ xJ ( P ) ⊆ P , ∀ x ∈ M . The J - real part of H is defined as H J := { ξ ∈ H : J ξ = ξ } and one has the decomposition H = H J ⊕ i H J . Moreover, one may define the positive part ξ + ∈ P of J -real vector ξ ∈ H J asthe Hilbert projection of ξ onto the positive cone P , its negative part ξ − ∈ P as the difference ξ − := ξ − ξ + and its modulus by | ξ | := ξ + + ξ − ∈ P so that ξ = ξ + − ξ − and ( ξ + | ξ − ) = 0.6.0.1. Standard form of commutative von Neumann algebras.
One may readily checks that( L ∞ ( X, m ) , L ( X, m ) , L ( X, m ) , J ) is a standard form of the commutative von Neumannalgebra L ∞ ( X, m ) (once the anti-unitary involution is given by the complex conjugation:
J a = a ) and that the above notions related to the order structure assume the familiarmeaning.6.0.2. Hilbert-Schmidt standard form.
A noncommutative example is provided by Hilbert-Schmidt standard form ( B ( h ) , L ( h ) , L ( h ) , J )of the algebra B ( h ) of all bounded operators on a Hilbert space h . In the Hilbert space L ( h )of all Hilbert-Schmidt operators on h , the cone L ( h ) of the positive ones is self-polar andthe involution J associates to the Hilbert-Schmidt operator a its adjoint a ∗ .Essential properties of the standard form of a von Neumann algebra are its existence anduniqueness (modulo unitaries preserving the positive cones). These properties authorize todenote the standard form of a von Neumann algebra M simply by( M, L ( M ) , L ( M ) , J ) . These main results are also of pratical use because different standard forms may show differentadvantages (or inconveniences). In the commutative case uniqueness is a reflection of the factthat the von Neumann algebra L ∞ ( X, m ) is determined by the class of zero m -measure setsonly, so that the algebra can be represented standardly on the space L ( X, m ′ ) of any measure m ′ equivalent to m .6.0.3. Standard form of semifinite von Neumann algebras.
In case the von Neumann algebras M is semifinite, a standard form may be constructed by the GNS representation associatedto a normal, semifinite trace τ on M as( M, L ( M, τ ) , L ( M, τ ) , J τ ) . To construct the standard form of a von Neumann algebra M starting from a normal state ω ∈ M ∗ + one need to recall some aspects of the Tomita-Takesaki Modular Theory of vonNeumann algebras [T1,2]. We may assume that M is represented in a Hilbert space H so that M ⊆ B ( H ) and ω is represented by a cyclic and separating vector ξ ∈ h as ω ( x ) = ( ξ | xξ ) H for x ∈ M (for example, H can be assumed to be the GNS space L ( M, ω )). The anti-linearmap S ( xξ ) := x ∗ ξ , densely defined on M ξ ⊆ H , is a closable operator on H and we mayconsider the polar decomposition of its closure ¯ S ¯ S = J ∆ / where the square root of the self-.adjoint modular operator ∆ = ¯ S ∗ ¯ S provides its positivepart and the modular conjugation J is an anti-unitary operator on H providing the phase.Using these tools one proves that P := { xJ xJ ∈ H : x ∈ M } is a self-polar cone in H coinciding with ∆ / M + ξ and that ( M, H , P , J ) is a standard form.When ω is a trace state then S is isometric so that the modular operator ∆ reduces to the identity, S = J and P = M + ξ . The modular operator ∆ measures how much the state ω differs from a trace state in that only in this case ∆ reduces to the identity.The denomination modular used for the operator ∆ originates from the following example.6.0.4. Modular operator and standard form of group von Neumann algebra.
Let (
G, m H ) be alocally compact group and consider the convolution algebra C c ( G ) acting by left convolution λ G on L ( G, m H ) and define the group von Neumann algebra L ( G ) as λ G ( C c ( G )) ′′ . TheHaar measure m H determines an additive, homogeneous, lower semicontinuous functional ω H on the positive part L ( G ) + , called the Plancherel weight (see [T3 Chapter VII]). It isa trace if and only if G is unimodular and a trace state if and only if G is discrete. Sinceon L ( G ) the involution is determined by λ G ( a ) ∗ := λ G ( a ∗ ) where a ∗ ( s ) := a ( s − ) for s ∈ G and a ∈ C c ( G ), one may check that the modular operator ∆ H on L ( G, m H ) associated tothe Plancherel weight ω H is given by the multiplication operator by the modular function G ∋ s dm H ( · s − ) /dm H .6.0.5. Modular operators and Gibbs states.
On the von Neumann algebra B ( h ) any normalstate ω can be represented by a self-adjoint, positive, compact operator ρ ∈ B ( h ) havingunit trace, called density matrix , as follows ω ( x ) = Tr ( xρ ) x ∈ B ( h ) . setting H := − ln ρ we have ρ = e − H so that ω ρ appears as the Gibbs equilibrium state ofthe dynamical system whose time evolution is given by the automorphisms group α t ( x ) = e − itH xe + itH x ∈ B ( h )generated by the Hamiltonian H . We now use the Hilbert-Schmidt standard form of B ( h )to compute the action of the modular operator. Since ω ( x ) = Tr ( xρ ) = Tr ( ρ / xρ / ) =( ρ / | xρ / ) L ( h ) we have that compact operator ξ = ρ / ∈ L ( h ) in the Hilbert-Schmidtclass is the cyclic and separating vector representing ω . To recover the action of the mod-ular operator notice that, by definition, we have J ∆ / ( xξ ρ ) = x ∗ ξ ρ for x ∈ B ( h ). Then∆ / ( xρ / ) = J ( x ∗ ρ / ) = ρ / x for all x ∈ B ( h ) so that∆ / ξ = ρ / ξρ − / ξ ∈ D (∆ / ) := { η ∈ L ( h ) : ρ / ηρ − / ∈ L ( h ) } . Notice that ( xJ xJ )( ξ ) = xξx ∗ for all x ∈ B ( h ) and ξ ∈ L ( h ) so that P = L ( h ).Another crucial property of the standard form is that any normal state ω ∈ M ∗ + can berepresented as the vector state of a unique, unit vector ξ ω ∈ P in the standard positive cone,i.e. ω ( x ) = ( ξ ω | xξ ω ) H for all x ∈ M .6.1. Tomita-Takesaki Theory and Connes’ Radon-Nikodym Theorem. ([T1,2], [Co2]).Let (
M, ω ) be a von Neumann algebra with a faithful, normal state and denote by π ω : M → B ( L ( M, ω )) the associated GNS representation. The Tomita-Takesaki Theorem then ensurethat J ω π ω ( M ) J ω = π ω ( M ) ′ , ∆ − itω π ω ( M )∆ itω = π ω ( M ) t ∈ R . Moreover, setting σ ωt : M → M σ ωt ( x ) := π − ω (∆ − itω π ω ( x )∆ itω ) x ∈ M, t ∈ R one gets a w ∗ -continuous group σ ω ∈ Aut( M ) of automorphisms of the von Neumann algebrathat satisfies and it is uniquely determined by the following modular condition ω ( xσ ω − i ( y )) = ω ( yx ) for all x, y ∈ M which are analytic with respect to σ ω . A fundamental theorem due to A.Connes [Co0 Theorem 1.2.1], which has to be considered as the noncommutative generaliza-tion of the Radon-Nikodym Theorem, states that the modular automorphism group of a vonNeumann algebra is essentially unique: for any pair φ, ψ ∈ M ∗ + of faithful, normal states on M , there exists a canonical 1-cocycle u : R → U ( M ) for σ φt ,with values in the unitary groupof M u t + t = u t σ φt ( u t ) t , t ∈ R , such that σ ψt ( x ) = u t σ φt ( x ) u ∗ t x ∈ M, t ∈ R . Modular operators on type I factors.
In case of the von Neumann algebra B ( h ) and thenormal state ω ( x ) := Tr( ρx ) associated to a positive, trace-class operator ρ ∈ B ( h ) with unittrace, one checks that the modular group is given by σ ωt ( x ) = ρ it xρ − it , x ∈ B ( h ) , t ∈ R and that the modular condition follows from the trace property of Tr ω ( yx ) = Tr( ρyx ) = Tr( ρxρyρ − ) = ω ( xσ ω − i ( y )) . In the particular case of a matrix algebra M n ( C ), denoting by e jk the matrix units, if thedensity matrix ρ is diagonal with eigenvalues λ , · · · , λ n >
0, one has σ ωt ( e jk ) = (cid:16) λ j λ k (cid:17) it e jk j, k = 1 , · · · , n, t ∈ R . Symmetric embeddings. ([Ara], [Cip1]). In the commutative case, the standardform of a probability space (
X, m ), we have the natural embeddings L ∞ ( X, m ) ⊆ L ( X, m ), L ( X, m ) ⊆ L ( X, m ) and L ∞ ( X, m ) ⊆ L ( X, m ).These may be generalized to the standard form of any von Neumann algebra M , using themodular operators associated to any fixed faithful normal state ω ∈ M ∗ + . Definition 6.2. (Symmetric embeddings) The symmetric embeddings associated to the stan-dard form ( M, H , P , J ) and a cyclic and separating vector ξ ω ∈ P are defined as follows:i) i ω : M → H i ω ( x ) := ∆ / ω xξ ω , x ∈ M ;ii) i ω ∗ : H → M ∗ h i ω ∗ ( ξ ) , y i = ( i ω ( y ∗ ) | ξ ) = (∆ / ω y ∗ ξ ω | ξ ) , ξ ∈ H , y ∈ M ;iii) j ω : M → M ∗ h j ω ( x ) , y i = ( J ω yξ ω | xξ ω ) , x, y ∈ M .These maps are well defined because M ξ ω ⊆ D (∆ / ω ), by the very definition of the modularoperator, and because D (∆ / ω ) ⊆ D (∆ / ω ) by the Spectral Theorem.The essential feature of these embeddings is that they preserve the order structures of M , H and M ∗ provided by the positive cones of these spaces. In particular i ω establishes a one toone homeomorphism between the set [0 , M ] := { x = x ∗ ∈ M : 0 ≤ x ≤ M } ⊂ M + and itsimage i ω ([0 , M ]) = { ξ ∈ P : 0 ≤ ξ ≤ ξ ω } := [0 , ξ ω ] ⊂ P .In the following we will indicate by ξ ∧ ξ the Hilbert projection onto the closed, convex set { ξ ∈ H J : ξ ≤ ξ ω } of a J -real vector ξ ∈ H J . In the commutative case and when ξ ω isgiven by the constant function 1 and a ∈ L R ( X, m ) is a real function, then a ∧ unit contraction of a given by ( a ∧ x ) = inf( a ( x ) ,
1) for x ∈ X . Using thisgeometric operation we may rephrase on the standard form of any von Neumann algebra M ,the Markovianity of Dirichlet forms one considers in the commutative setting L ∞ ( X, m ). Definition 6.3. (Dirichlet forms [Cip1]). Let ( M, H , P , J ) be a standard form of a vonNeumann algebra M and ξ ω ∈ P a cyclic and separating vector representing the normal state ω ∈ M ∗ + .A quadratic form E : H → ( −∞ , + ∞ ] is said to be J -real if E [ J ξ ] = E [ ξ ] ξ ∈ H and Markovian with respect to ξ ω if it is J -real and(6.1) E [ ξ ∧ ξ ω ] ≤ E [ ξ ] ∀ ξ ∈ H J . In case E [ ξ ω ] = 0, the Markovianity condition is equivalent to E ( ξ + | ξ − ) ≤ ξ = J ξ ∈ H . A densely defined, lower semicontinuous Markovian form is called a Dirichlet form withrespect to ξ ω or ω .The quadratic form E is called a completely Dirichlet form if any of its matrix extension E n on H ⊗ L ( M n ( C ), given by E n [[ a i,j ] ni,j =1 ] = n X i,j =1 E [ a i,j ] [ a i,j ] ni,j =1 ∈ H ⊗ L ( M n ( C ) , is a Dirichlet form w.r.t. the state ω ⊗ τ n , where τ n is the unique tracial state on M n ( C ).In particular, if E is Markovian and E [ ξ ] is finite then E [ ξ ∧ ξ ω ] is finite too. Also, in thecommutative setting Dirichlet forms are automatically completely Dirichlet forms. In otherwords, under the Hilbertian projection ξ ξ ∧ ξ ω , the value of the quadratic form does notincrease. As noticed above, this definition reduces to the usual one in the commutative setting.We are going to see that in any standard form, Dirichlet forms represent an infinitesimalcharacterization of strongly continuous, symmetric Markovian semigroups. Theorem 6.4. (Characterization of Markovian semigroups by Dirichlet forms [GL1,2], [Cip1].)Let ( M , H , P J ) be a standard form of a von Neumann algebra M and ξ ω ∈ P the cyclicvector representing a state ω ∈ M ∗ + . Let { T t : t ≥ } be a J -real, symmetric, strongly contin-uous, semigroup on the Hilbert space H and E : H → ( −∞ , + ∞ ] the associated J -real, lowersemibounded, closed quadratic form. Then, the following properties are equivalenti) { T t : t ≥ } is Markovian with respect to ξ ω ;ii) E is a Dirichlet form with respect to ξ ω .In particular, Dirichlet forms are automatically nonnegative and Markovian semigroups areautomatically contractive. Dirichlet forms not only determine and are determined by strongly continuous Markoviansemigroups on the standard Hilbert space, but they are also in one-to-one correspondencewith point-weak*-continuous, completely positive, subunital (abbreviated with Markovian)semigroups on the von Neumann algebra, satisfying a certain modular symmetry property which is a deformation of the modular condition.
Theorem 6.5. (Markovian semigroups on standard forms of von Neumann algebras [Cip1]).Let ( M , H , P J ) be a standard form of a von Neumann algebra M , ω ∈ M ∗ + a faithful stateand ξ ω ∈ P its representing cyclic vector. Then there exists a one-to-one correspondencebetweeni) Markovian (with respect to ξ ω ) semigroups { T t : t > } on L ( M ) and ii) Markovian semigroups { S t : t > } on M which are ω -modular symmetric in the sensethat, for all x, y in a weak*-dense, σ ω -invariant , ∗ -subalgebra of M σ ω , one has (6.2) ω (cid:16) yS t ( x ) (cid:17) = ω (cid:16) σ ω i ( x ) S t ( σ ω − i ( y )) (cid:17) . The correspondence is provided by the symmetric embedding through the relation i ω : M → L ( M ) i ω ◦ S t = T t ◦ i ω . Remark . A careful analysis of the family of closed cones { ∆ αω xξ ω : x ∈ M + } in L ( M )for α ∈ (0 , / GNSembedding x xξ ω of M into L ( M ) is used, the resulting semigroup automatically commuteswith modular operator. In application to convergence to equilibrium in Quantum StatisticalMechanics this situation should be avoided and this is the reason why the self-polar conecorresponding to α = 1 / Elementary Dirichlet forms.
As a first example of a Dirichlet form with respect to anot necessarily trace state, we illustrate a construction that can be considered a generalizationof the one of Albeverio-Hoegh-Khron in Theorem 4.5 above. Elementary Dirichlet form willfind application to approximation/rigidity properties of von Neumann algebras in Section 7.6.Let ( M , H , P J ) be a standard form and ξ ∈ P a cyclic vector. Consider, for fixed a k ∈ M , µ k , ν k > k = 1 , . . . , n , the operators ∂ k : H → H d k := i ( µ k a k − ν k j ( a ∗ k ))and the bounded quadratic form E [ ξ ] := n X k =1 k ∂ k ξ k ξ ∈ H . Then E is J -real and E ( ξ + | ξ − ) ≤ J -real ξ ∈ H if and only if n X k =1 [ µ k a ∗ k a k − ν k a k a ∗ k ] ∈ M ∩ M ′ . Moreover, if the above condition holds true, E is a Dirichlet form with E [ ξ ] = 0 if andonly if the numbers ( µ k /ν k ) , k = 1 , . . . , n , are eigenvalues of the modular operator ∆ ξ ,corresponding the eigenvectors a k ξ ∈ H . Conditions like the one above are considered in theframework of q-deformed CCR relations and related factor von Neumann algebras [Boz].The construction above provides a, possibly unbounded, Dirichlet form even when n = ∞ ,provided E is densely defined.6.2.2. Quantum Ornstein-Uhlenbeck and Quantum Brownian motion semigroups. ([CFL]).We describe here the construction of a Dirichlet form, on the Neumann algebra B ( h ), whichgenerates a Markovian semigroups appearing in quantum optics.On the Hilbert space h := l ( N ) consider the standard form ( B ( h ) , L ( h ) , L ( h ) , J ). Let { e n : n ≥ } ⊂ h be the canonical Hilbert basis, and denote by | e m ih e n | , n, m ≥
0, thepartial isometries, having C e n as initial space and C e m as final one.Fix the parameters µ > λ >
0, set ν := λ /µ and let ω ν ( x ) := Tr( ρ ν x ) the normal state on B ( h ) represented by the density matrix ρ ν := (1 − ν ) X n ≥ ν n | e n ih e n | . The state ω ν is then represented by the cyclic vector ξ ν := ρ / ν = (1 − ν ) / P n ≥ ν n/ | e n ih e n | .The creation and annihilation operators , a ∗ and a on h , are defined by a ∗ e n := √ n + 1 e n +1 , ae n := (cid:26) √ ne n − , if n > , if n = 0.They are adjoint of one another on their common domain D ( a ) = D ( a ∗ ) = (cid:8) e ∈ h : P n ≥ √ n |h e | e n i| < ∞ (cid:9) and satisfy the CCR a a ∗ − a ∗ a = I .
The quadratic form given by E [ ξ ] := k µaξ − λξa k + k µaξ ∗ − λξ ∗ a k , densely defined in L ( h ) on the subspace of finite rank operators D ( E ) := linear span {| e m ih e n | , n, m ≥ } , is closable and Markovian so that its closure is a Dirichlet form with respect to ω ν , generatingthe so called quantum Ornstein-Uhlenbeck Markovian semigroup. Moreover, since, as it is easyto check one has E [ ξ ν ] = 0, it results that the cyclic vector is left invariant by the semigroup.When λ = µ , the role of the invariant state ω ν has to be played by the normal, semifinitetrace τ on B ( h ). However, even in this case, using the Albeverio-Hoegh-Khron criterion, itis possible to prove that the closure of the unbounded quadratic form E [ ξ ] := k aξ − ξa k + k aξ ∗ − ξ ∗ a k , ξ ∈ D ( E )is a Dirichlet form. The associated τ -symmetric Markovian semigroup on B ( h ), may bedilated by a Quantum Stochastic Process, known as the Quantum Brownian motion . Thisrepresents a sort of bridge between pairs of classical stochastic processes of quite differenttype. In fact on a suitable, invariant, maximal abelian subalgebra (masa), this semigroupreduces to the semigroup of a classical Brownian motion while on another masa, it reducesto the semigroup of a classical birth and death process.7.
Application to approximation/rigidity propertiesof von Neumann algebras
In this section we describe three results showing that the spectral properties of Dirichletforms are naturally and deeply connected with those fundamental properties of von Neumannalgebras having to do with the ideas of approximation and rigidity.7.1.
Amenable groups.
In 1929 J. von Neumann discovered a far reaching explanation ofthe Banach-Tarski paradox in terms of a property, called amenability , of a group of Euclideanmotions in R n which holds true in dimension n = 1 , Definition 7.1.
A discrete group Γ is amenable if there exist a left-translation invariantprobability measure on Γ.This property is equivalent to the existence of a sequence of finitely supported, positivedefinite functions φ n on Γ, pointwise converging to the constant function 1,lim n φ n ( t ) = 1 for all t ∈ Γ , and to the existence of a proper , conditionally negative definite function ψ : Γ → C (see5.1.1). Recall that a function φ : Γ → C is positive definite if the matrices [ φ ( s − j s k )] nj,k =1 arepositive definite for all s · · · , s k ∈ Γ, i.e. if for all c , · · · c n ∈ C one has n X j,k =1 ¯ c j φ ( s − j s k ) c k ≥ . Since positive definite functions are coefficients of unitary representations and the constantfunction 1 is the coefficient of the trivial representation, amenability is also equivalent tothe fact that the trivial representation is weakly contained in the left regular one, i.e. it isunitarily equivalent to a subrepresentation of a multiple of the regular representation.The amenability of a group Γ can be translated in terms of a corresponding property of itsassociated group von Neumann algebra L (Γ).To introduce this property in complete generality, we recall the fundamental notions of bi-module and correspondence.7.2. Bimodules and Connes correspondences. ([Po], [Co2], [AP]). A Banach M - bimodule E on a C ∗ -algebra M is a Banach space E together with a pair of norm continuous, commutingactions of M .If the left action of x ∈ M on ξ ∈ E is denoted by xξ and the right action of y ∈ M on ξ ∈ E is denoted by ξy , the required commutation reads ( xξ ) y = x ( ξy ).In case M is a von Neumann algebra and E is a dual bimodule, in the sense that it is thedual Banach space of a predual one, the left and right actions are required to be continuouswith respect to the weak*-topology of E .A Connes correspondence on M is a Hilbert space H which is an M -bimodule.Denote by M ◦ the opposite algebra of M : it coincides with M as a vector space but theproduct is taken in the reverse order x ◦ y ◦ := ( yx ) ◦ for x ◦ , y ◦ ∈ M ◦ . By convention, elements y ∈ M , when regarded as elements of the opposite algebra are denoted by y ◦ ∈ M ◦ . Let M ⊗ max M ◦ the maximal tensor product of M and M ◦ considered as C ∗ -algebras.A correspondence on M is nothing but a representation π : M ⊗ max M ◦ → B ( H ) π ( x ⊗ y ◦ ) ξ = xξy such that M ∋ x π ( x ⊗ M ) and M ∋ x π (1 M ⊗ x ◦ ) provide normal representa-tions. Correspondences of von Neumann algebras may be conveniently thought of both asgeneralization of unitary representations of groups.Among the correspondences of von Neumann algebras, the following are of central importance.The identity or standard correspondence of a von Neumann algebra M is provided by itsstandard representation ( M, L ( M ) , L ( M ) , J ). Here beside the left action of M on L ( M ),denoted by xξ for x ∈ M and ξ ∈ L ( M ), we have the right action defined by ξx := J x ∗ J ξ .The coarse correspondence is the M -bimodule given by L ( M ) ⊗ L ( M ) with actions x ( ξ ⊗ η ) y = xξ ⊗ ηy x, y ∈ M, ξ, η ∈ L ( M ) . This is also called the
Hilbert-Schmidt correspondence by the identification of L ( M ) ⊗ L ( M )with the Hilbert space HS ( L ( M )) of Hilbert-Schmidt operator on L ( M ). In this terms theactions are given by xT y ∈ HS ( L ( M )) for x, y ∈ M and T ∈ HS ( L ( M )). Correspondences of von Neumann algebras may also be fruitfully thought as generalization ofcompletely positive maps. In fact, suppose that on M a faithful, normal state ω is fixed andconsider a not necessarily self-adjoint, completely Markovian map T : L ( M, ω ) → L ( M, ω ),assuming, to simplify, that
T ξ ω = ξ ω . Then the functional determined byΦ T : M ⊗ max M ◦ → C Φ T ( x ⊗ y ◦ ) := ( i ω ( y ∗ ) | T i ω ( x ))is a state on M ⊗ max M ◦ which, by the GNS construction, give rise to a representation of M ⊗ max M ◦ , thus to a correspondence H T on M . The unit, cyclic vector ξ T ∈ H T representingΦ t thus satisfies ( i ω ( y ∗ ) | T i ω ( x )) = ( ξ T | xξ T y ) H T x, y ∈ M. A fundamental operation that is defined on correspondences is their relative tensor product ,by which any M - N -correspondence H N and any N - P -correspondence K P may tensorized,in this order, to produce an M - P -correspondence denoted by H ⊗ K P . The advantagesto translate into the common language of correspondences problems of apparently differentorigin concerning von Neumann algebras, are the possibility to let them play into a sharedground on one side, and the possibility to use the tools of representation theory, for exampleto introduce notions like containment, weak containment and convergence.7.3. Amenable von Neumann algebras.Definition 7.2. ([Co2,3,4], [CE]). A C ∗ or von Neumann algebra M is said to be amenable if for every dual Banach M -bimodule E , all derivations δ : M → X , that is maps satisfyingthe Leibniz property δ ( ab ) = ( δa ) b + a ( δb ) a, b ∈ M, are inner, i.e. there exists ξ ∈ E such that δ ( x ) = xξ − ξx x ∈ M. This property was introduced by Johnson and Ringrose in their works on cohomology ofoperator algebras. As the result of an enormous amount of efforts, it has been shown thatamenability is equivalent to approximation properties:i) a C ∗ -algebra A is amenable if and only if it is nuclear in the sense that its identity mapcan be approximated in the point-norm topology,lim n k ψ n ◦ φ n ( a ) − a k = 0 for all a ∈ A, by the composition of suitable contractive, completely positive maps ψ n : A → M k n ( C ) φ n : M k n ( C ) → A ;ii) a von Neumann algebra M is weakly nuclear if and only if its identity map can be approx-imated in the point-ultraweak topology,lim n η ( ψ n ◦ φ n ( a ) − a ) = 0 for all a ∈ A, η ∈ M ∗ , by the composition of suitable contractive, completely positive maps ψ n : A → M k n ( C ) φ n : M k n ( C ) → A ;iii) a von Neumann algebra M is amenable if and only if it is hyperfinite in the sense that itis generated by an increasing sequence of finite-dimensional subalgebras.Among the examples of amenable von Neumann algebras, one may recall i) the group vonNeumann algebra of a locally compact amenable group ii) the crossed product of an abelian von Neumann algebra by an amenable locally compact group iii) the commutant von Neu-mann algebra of any continuous unitary representation of a connected locally compact groupand iv) the von Neumann algebra generated by any representation of a nuclear C ∗ -algebra.7.4. Amenability and subexponential spectral growth rate of Dirichlet forms. ([CS5]). To illustrate a first connection between approximation properties of von Neumannalgebras and spectral properties of Dirichlet form, we first recall a definition.
Definition 7.3. (Spectral growth rate of Dirichlet forms [CS5]). Let (
N, ω ) be an infinitedimensional, σ -finite, von Neumann algebra with a fixed faithful, normal state on it.Let ( E , F ) be a Dirichlet form on L ( N, ω ) and let (
L, D ( L )) be the associated nonnegative,self-adjoint operator. Assume that its spectrum σ ( L ) = { λ k ≥ k ∈ N } is discrete , setΛ n := { k ∈ N : λ k ∈ [0 , n ] } , β n := ♯ (Λ n ) , n ∈ N and define the spectral growth rate of ( E , F ) asΩ( E , F ) := lim sup n ∈ N n p β n . The Dirichlet form ( E , F ) is said to have exponential growth if ( E , F ) has discrete spectrum and Ω( E , F ) > subexponential growth if ( E , F ) has discrete spectrum and Ω( E , F ) = 1 polynomial growth if ( E , F ) has discrete spectrum and, for some c, d > n ∈ N , β n ≤ c · n d . intermediate growth if it has subexponential growth but not polynomial growth.It is easy to see that the subexponential growth property can be formulated in terms of nuclearity of the Markovian semigroup { e − tL : t > } on L ( N, ω ): Lemma 7.4.
The Dirichlet form ( E , F ) has discrete spectrum and subexponential spectralgrowth rate if and only if the Markovian semigroup { e − tL : t > } on L ( N, ω ) is nuclear, ortrace-class, in the sense that: Trace ( e − tL ) < + ∞ t > . Here is the announced connection between amenability and spectral properties.
Theorem 7.5. ([CS5]). Let N be a σ -finite von Neumann algebra. If there exists a normal,faithful state ω ∈ M ∗ + and a Dirichlet form ( E , F ) on L ( N, ω ) having subexponential spectralgrowth, then N is amenable. Let us sketch the main points of the proof assuming, to simplify, that E [ ξ ω ] = 0. Let N ⊗ N ◦ the von Neumann spatial tensor product of N and N ◦ . It turns out that the coarse represen-tation of N ⊗ max N ◦ , defined by π co : N ⊗ max N ◦ → B ( L ( N, ω ) ⊗ L ( N, ω )) π co ( x ⊗ y o )( ξ ⊗ η ) := xξ ⊗ ηy x, y ∈ N , ξ, η ∈ L ( N, ω ) , give rise to the spatial tensor product of the von Neumann algebras( π co ( N ⊗ max N ◦ )) ′′ = N ⊗ N ◦ . Moreover, the normal extension of the coarse representation π co of N ⊗ max N ◦ to N ⊗ N ◦ isthe standard representation of N ⊗ N ◦ so that L ( N, ω ) ⊗ L ( N, ω ) ≃ L ( N ⊗ N ◦ , ω ⊗ ω ◦ )and the positive cone L ( N ⊗ N ◦ , ω ⊗ ω ◦ ) can be identified with the cone of all positive,Hilbert-Schmidt operators on L ( N, ω ). In particular, since, by assumption, e − tL is a positive,Hilbert-Schmidt operator for all t >
0, we have e − tL ∈ L ( N ⊗ N ◦ , ω ⊗ ω ◦ ) t > . Since E is a complete Dirichlet form, its associated semigroup is completely positive and thisimplies that the linear functional Φ t , determined byΦ t : N ⊗ max N ◦ → C Φ t ( x ⊗ y ◦ ) := ( i ω ( y ∗ ) | e − tL i ω ( x )) , is positive and actually a state since E [ ξ ω ] = 0. By the continuity properties of the symmetricembeddings and the above identifications, we haveΦ t ( z ) = (cid:16) e − tL (cid:12)(cid:12)(cid:12) i ω ⊗ ω ◦ ( z ) (cid:17) L ( N ⊗ N ◦ ,ω ⊗ ω ◦ ) z ∈ N ⊗ N ◦ . Since i ω ⊗ ω ◦ is positive preserving and e − tL is a positive element of the standard cone, we havethat Φ t is a normal state on N ⊗ N ◦ and can thus be represented by a unique positive unitvector Ω t ∈ L ( N ⊗ N ◦ , ω ⊗ ω ◦ ) asΦ t ( z ) = (cid:16) Ω t | π co ( z )Ω t (cid:17) L ( N ⊗ N ◦ ,ω ⊗ ω ◦ ) z ∈ N ⊗ N ◦ . By the strong continuity of the Markovian semigroup e − tL on L ( N, ω ), we then havelim t ↓ (cid:16) Ω t | π co ( z )Ω t (cid:17) L ( N,ω ) ⊗ L ( N,ω ) = ( ξ ω | π id ( z ) ξ ω ) = 1 z ∈ N ⊗ max N ◦ . This proves that the trivial representation π id of N ⊗ max N ◦ , given by π id ( z ) := I L ( N,ω ) for all z ∈ N ⊗ max N ◦ , is weakly contained in the coarse representation π co and thus N is amenableby a characterization of amenability due to S. Popa [Po].This approach by correspondences to relate spectral properties of Dirichlet forms to ap-proximation properties of von Neumann algebras allows to treat also the relative case in whichone deals with embeddings of subfactors B ⊂ N on one side and with the a subexponentialspectral growth rate of Dirichlet form r elative to the subalgebra B , on the other side. In thesesituations the essential spectrum of Dirichlet forms is not empty. (See [CS5]).7.5. Haagerup approximation property and discrete spectrum of Dirichlet forms.
The free group of two generators F is non amenable but in 1979 U. Haagerup proved in [H3]that its word-length function l is conditionally negative definite and proper. This allowedhim to prove that the group von Neumann algebra L ( F ) and the group C ∗ -algebra of F have the Grothendieck Metric Approximation Property. Moreover, the above properties ofthe length function of free groups also determine the following properties. This specific caseopened the study of the following class of groups, larger than the class of amenable ones. Definition 7.6.
A countable, discrete group Γ is said to have the
Haagerup ApproximationProperty if there exists a sequence of positive definite functions in c (Γ), uniformly convergenton compact subsets, to the constant function 1 (see for example [CCJJV]). This property isequivalent to the existence of a proper, conditionally negative definite function on Γ. Clearly all amenable groups have the Property (H). A series of contribution [Ch], [CS1,2],[COST] [J], allowed to isolate the following property of von Neumann algebras that for groupalgebras L (Γ) of discrete groups is equivalent to the Haagerup Approximation Property of Γ. Definition 7.7.
A von Neumann algebra with standard form (
M, L ( M ) , L ( M ) , J ) is saidto have the Haagerup Approximation Property (HAP) if there exists a sequence of completelypositive contractions T k : L ( M ) → L ( M ) strongly converging to the identity operatorlim k k ξ − T k ξ k L ( M ) = 0 ξ ∈ L ( M ) . Here is the announced connection between (HAP) and spectral properties.
Theorem 7.8. ([CS1]). Let N be a σ -finite von Neumann algebra. Then the followingproperties are equivalenti) M has the Property (HAP)ii) there exists on L ( M ) a completely Dirichlet form ( E , F ) with respect to some faithful,normal state ω ∈ M ∗ + , having discrete spectrum.Remark . i) Property (H) may be formulated in a number of slightly different, equivalentways also for not necessarily σ -finite von Neumann algebras too in such a way that the abovespectral characterization remains anyway true.ii) The construction of Dirichlet forms out of negative definite functions on groups and theabove characterization of the Haagerup Approximation Property, indicate that Dirichlet formsfor arbitrary von Neumann algebras play a role parallel to the one played by the continuous,negative definite functions on groups (see discussion in [CS1]).7.6. Property ( Γ ) and Poincar´e inequality for elementary Dirichlet forms. Anotherinstance of the interactions among structural properties of a von Neumann algebra M andspectral properties of Dirichlet forms on L ( M ) may be shown reformulating the Murray-vonNeumann Property (Γ).By an elementary completely Dirichlet form on a finite von Neumann algebra ( M, τ ), endowedwith a normal, trace state, we mean one of type E F [ ξ ] := X x ∈ F k xξ − ξx k L ( M,τ ) ξ ∈ L ( M, τ )for some finite, symmetric set F = F ∗ ⊂ M . The unit, cyclic vector ξ τ ∈ L ( M, τ ) repre-senting the trace is central so that E F [ ξ τ ] = 0 and λ = 0 is an eigenvalue for all elementaryDirichlet forms. Elementary Dirichlet forms are everywhere defined and thus bounded. Definition 7.10. ([Co1]). A finite von Neumann algebra endowed with its normal, tracialstate (
M, τ ), has the Property (Γ) if for any ε > F ⊂ M there exists aunitary u ∈ M with τ ( u ) = 0 such that k ( ux − xu ) ξ τ k < ε for all x ∈ F .This property was the first invariant introduced by F.J. Murray and J. von Neumann [MvN]to show the existence of non hyperfinite II factors. For example, the group von Neumannalgebra L ( S ∞ ) of the countable discrete group S ∞ of finite permutations of a countable setand the Clifford von Neumann algebra of a separable Hilbert space are both isomorphic tothe hyperfinite II factor R , which fullfill the Property (Γ). This latter cannot be isomorphicto the group algebra L ( F n ) of the free group F n with n ≥ II factorbut does not have the Property (Γ) (and in fact it is not hyperfinite). It is well known [Po] that the absence of the Property (Γ) for a II factor with separable pred-ual, is a rigidity property equivalent to the existence of a spectral gap for suitable self-adjoint,finite, convex combinations of inner automorphisms, as unitary operators on L ( M, τ ).We now show how the Property (Γ) can be also naturally interpreted in terms of a spectralproperty of elementary Dirichlet forms.
Theorem 7.11. ([CS6]). A finite von Neumann algebra endowed with its normal, tracialstate ( M, τ ) , has the Property (Γ) if and only if for any elementary completely Dirichlet form E F [ ξ ] = X x ∈ F k xξ − ξx k L ( M,τ ) ξ ∈ L ( M, τ ) , associated to a finite set F = F ∗ ⊂ M , the eigenvalue λ := 0 is not isolated in the spectrum.Otherwise stated, ( M, τ ) , does not have the Property (Γ) if and only if there exists an el-ementary Dirichlet form E F such that the eigenvalue λ = 0 is isolated (spectral gap) or,equivalently, E F satisfies, for some c F > , a Poincar´e inequality c F · k ξ − ( ξ τ | ξ ) ξ τ k ≤ E F [ ξ ] ξ ∈ L ( M, τ ) . Proof. If J denotes the symmetry on L ( M, τ ) which extends the involution of M , then for u, x ∈ M , setting ξ := xξ τ ∈ M ξ τ , we have ( ux − xu ) ξ τ = uξ − J u ∗ x ∗ ξ τ = uξ − J u ∗ J xξ τ = uξ − ξu . Since ξ τ ∈ L ( M, τ ) is cyclic, i.e.
M ξ τ is dense in L ( M, τ ), if (
M, τ ) does nothave the Property Γ there exists ε > E F such that for allunitaries u ∈ M we have ε · k uξ τ − τ ( u ) ξ τ k ≤ E F [ uξ τ ] . For any y = y ∗ ∈ M such that k y k M ≤ / √
2, consider the unitaries u ± := y ± i p M − y so that y = ( u + + u − ) /
2. Since p − y = φ ( y ) with φ ( s ) := √ − s and | φ ′ ( s ) | ≤ | s | ≤ / √
2, it follows by the Markovianity of the Dirichlet form that E F [ p − y ξ τ ] ≤ E F [ yξ τ ].Since E F is J -real we then have E F [ u ± ξ τ ] = E F [( y ± i p − y ) ξ τ ] = E F [ yξ τ ] + E F [ p − y ξ τ ] ≤ E F [ yξ τ ]and ε · ( k u + ξ τ − τ ( u + ) ξ τ k + k u − ξ τ − τ ( u − ) ξ τ k ) ≤ E F [ u + ξ τ ] + E F [ u − ξ τ ] ≤ E F [ yξ τ ] . Thus for all y ∈ M we have k yξ τ − τ ( y ) ξ τ k = k u + ξ τ − τ ( u + ) ξ τ + u − ξ τ − τ ( u − ) ξ τ k ≤ (cid:0) k u + ξ τ − τ ( u + ) ξ τ k + k u − ξ τ − τ ( u − ) ξ τ k (cid:1) ≤ ε − · E F [ yξ τ ] . and, by the density of M ξ τ in L ( M, τ ), a Poincar´e inequality holds true with c F = ε/ (cid:3) By classical results of A. Connes [Co3], obtained along his classification of injective factors,one can relate the existence of spectral gap for an elementary Dirichlet form to fundamentalproperties of II factors ( M, τ ) with separable predual: the following properties are equivalenti) the subgroup Inn( M ) of inner automorphisms is closed in Aut( M ) ( M is called a full factor )ii) the C ∗ -algebra C ∗ ( M, M ′ ) generated by M and its commutant M ′ , acting standardly on L ( M, τ ), contains the ideal of compact operatorsiii) there exists an elementary Dirichlet form E F on L ( M, τ ) satisfying a Poincar´e inequality.
Property (T).
Groups having the Kazdhan property (T) show, in many instances, avery rigid character. By their definition, all continuous, negative definite functions on themare bounded (see [CCJJV]) and they can be characterized by any of the following properties:i) whenever a sequence of continuous, positive definite functions converges to 1 uniformlyon compact subsets, then it converges uniformly ii) if a representation contains the trivialrepresentation weakly, then it contains it strongly iii) every continuous, isometric action onan affine Hilbert space has a fixed point.In von Neumann algebra theory, the Property (T) of a group Γ with infinite conjugacy classes,were first considered by A. Connes to show that the factor L (Γ) has a countable fundamentalgroup. The same author characterized countable, discrete groups Γ having the Property (T)through a specific property of L (Γ). Later A. Connes and V. Jones [CJ] identified a property(T) for general von Neumann algebras in strong analogy with one of the above character-izations for the groups case. They key point was the replacement of the notion of grouprepresentation by that of correspondence for general von Neumann algebras: M has the property (T) if all correspondences sufficiently close to the standard one mustcontain it .In Section 10 below, we will describe a recent result by A. Skalski and A. Viselter to a Dirich-let form characterization of the property (T) of von Neumann algebras of locally compactquantum groups. 8. KMS-symmetric semigroups on C ∗ -algebras We have seen that the extension of the theory of Dirichlet forms introduced by S. Albeverioand R. Hoegh-Khron and developed and applied by J.-L. Sauvageot [S3,7,8] and by E.B.Davies [D2], E.B. Davies and O. Rothaus [DR1,2] and by E.B. Davies and M. Lindsay [DL1,2],can be applied to several fields in which the relevant algebra of observables, to retain aphysical language, is no more commutative. This theory concerns, however, C ∗ -algebras orvon Neumann algebras endowed with a well behaved trace functional. To have a theorysuitable to be applied to other fields one has to face the problem to give a meaning toMarkovianity of Dirichlet forms with respect non tracial states. For example,i) equilibria in Quantum Statistical Mechanics or Quantum Field Theory are described bystates obeying the Kubo-Martin-Schwinger condition which are not trace at finite temperatureii) in Noncommutative Geometry the algebra generated by the ”coordinate functions” of anoncommutative space may have a natural relevant state which is not a trace, as it is thecase of the Haar state of several Compact Quantum Groups.In this section we describe this extension of the theory of Dirichlet forms which deals withMarkovianity with respect to KMS states on C ∗ -algebras and with any normal, faithful stateson von Neumann algebras. In the next sections we shall have occasion to describe applicationswere this generalized theory is due.8.1. KMS-states on C ∗ -algebras. Let A be a C ∗ -algebra and let { α t : t ∈ R } be a stronglycontinuous automorphism group on it, often interpreted as a dynamical system. Definition 8.1. ( KMS-states )([Kub], [HHW]). Let α := { α t : t ∈ R } be a strongly con-tinuous group of automorphisms of a C ∗ -algebra A and β ∈ R . A state ω is said to be a( α, β )-KMS state if it is α -invariant and if the following KMS-condition holds true: ω ( aα iβ ( b )) = ω ( ba )for all a, b in a norm dense, α -invariant ∗ -algebra of analytic element for α . If M is a vonNeumann algebra and α := { α t : t ∈ R } is a w ∗ -continuous group of automorphisms, a state ω is said to be a ( α, β )-KMS state if ω is α -invariant, normal and the KMS-conditionabove holds true for all a, b in a σ ( M, M ∗ )-dense, α -invariant ∗ -subalgebra of A α . KMS statescorresponding to β = 0 are just the traces over M .Notice that any faithful normal state ω on a von Neumann algebra M is a ( σ ω , − σ ω at inverse temperature β = −
1. In thiscase, in fact, the KMS condition coincides with modular condition.
Definition 8.2. ( KMS-symmetric Markovian semigroups on C ∗ -algebras )([Cip2,5]).Let α := { α t : t ∈ R } be a strongly continuous group of automorphisms of a C ∗ -algebra A and ω be a fixed ( α, β )-KMS state, for some β ∈ R .A bounded map R : A → A is said to be ( α, β )- KMS symmetric with respect to ω if(8.1) ω (cid:16) bR ( a ) (cid:17) = ω (cid:16) α − iβ ( a ) R ( α + iβ ( b )) (cid:17) for all a, b in a norm dense, α -invariant ∗ -algebra of analytic elements for α .A strongly continuous semigroup { R t : t ≥ } on A is said to be ( α, β )- KMS symmetric withrespect to ω if R t is ( α, β )-KMS symmetric with respect to ω for all t ≥ ω is assumed to be normal, maps and semigroups to bepoint-weak*-continuous and the subalgebra B to be weak*-dense.Let α := { α t : t ∈ R } be a strongly continuous group of automorphisms of a C ∗ -algebra A and ω be a fixed ( α, β )-KMS state, for some β ∈ R . Let ( π ω , H ω , ξ ω ) be the correspondingGNS-representation, b ω the normal extension of ω to the von Neumann algebra M := π ω ( A ) ′′ and b α := { b α t : t ∈ R } be the induced weak*-continuous group of automorphisms of M .Comparing the KMS condition for b ω with respect to b α to its modular condition, one readilyobserve that the modular group of b ω is given by σ b ωt = b α − βt t ∈ R . The following is a key consequence of the ( α, β )-KMS-symmetry of a map.
Lemma 8.3. ([Cip2]) A map R : A → A which is ( α, β ) -KMS symmetric with respect to ω ,leaves globally invariant the kernel ker( π ω ) of the GNS-representation of ω . This result allows to study KMS symmetric maps and semigroups on the von Neumannalgebra associated to the GNS representation of the KMS state.
Theorem 8.4. (von Neumann algebra extension of KMS-symmetric semigroups)([Cip2])Let { R t : t ≥ } be a strongly continuous semigroup on A , ( α, β ) -KMS symmetric withrespect to ω . Then there exists a unique point-weak*-continuous semigroup { S t : t ≥ } on M determined by (8.2) S t ( π ω ( a )) = π ω ( R t ( a )) , a ∈ A , t ≥ . This extension is b ω -modular symmetric in the sense that (8.3) ˆ ω (cid:16) S t ( x ) σ b ω − i ( y ) (cid:17) = b ω (cid:16) σ b ω + i ( x ) S t ( y ) (cid:17) t ≥ , for all x, y in a weak*-dense, σ b ω -invariant ∗ -algebra of analytic elements σ b ω . Moreover,if { R t : t ≥ } is positive, completely positive, Markovian or completely Markovian, then { S t : t ≥ } shares the same properties. As a consequence, a Dirichlet form on L ( A, ω ) is determined by the semigroup on A Corollary 8.5.
Let ( L, D ( L )) be the generator of the semigroup { R t : t ≥ } on A . Then theDirichlet form on L ( A, ω ) associated to the strongly continuous extension of the b ω -modularsymmetric semigroup { S t : t ≥ } on M , satisfies the relation E [ i ω ( π ω ( a ))] = ( i ω ( π ω ( a )) | i ω ( π ω ( La ))) L ( A,τ ) a ∈ D ( L ) . By this result one may study properties of the semigroup R on the C ∗ -algebra through theassociated Dirichlet form E on A . Notice that by definition we have the coincidence of thespaces L ( A, ω ) = L ( M, b ω ).This result suggests also that one can approach the construction of Markovian semigroups( α, β )-symmetric with respect to a ( α, β )-KMS state ω on a C ∗ -algebra A , through the con-struction of Dirichlet forms on L ( A, ω ). The advantage being that working with quadraticforms instead that linear operators often allows to relax domain constrains to prove closabil-ity. To finalize this approach, however, once obtained from the Dirichlet form on L ( A, τ )the Markovian semigroup on the von Neumann algebra L ∞ ( A, τ ), one has to face the prob-lem to show that the C ∗ -algebra A is left invariant and that on it the semigroup is not onlyw ∗ -continuous but in fact strongly continuous. This last problem may be solved case by caseas we did for the Ornstein-Uhlenbeck semigroup in [CFL] for example. We notice, however,that even in classical potential theory, on Riemmanian manifolds the construction of the heatsemigroup on the algebra of continuous functions requires a certain amount of substantialpotential analysis ([D2]).9. Application to Quantum Statistical Mechanics
After the proof, in the early nineties of the last century, by D. Stroock and B. Zegarlin-ski, of the equivalence between the Dobrushin-Shlosman mixing condition and the uniformlogarithmic Sobolev inequalities for classical spin systems with continuous spin space, effortswere directed to obtain for quantum spin systems similar results, within the framework ofthe studies of the convergence to equilibrium. See for example [LOZ], [MZ1], [MZ2], [MOZ],[Mat1], [Mat2], [Mat3].In this section we describe just one of these constructions of Markovian semigroups by Dirich-let forms for KMS states of quantum spin systems, provided by Y.M. Park and his school[P1], [P2], [P3], [BKP1], [BKP2].9.1.
Heisenberg Quantum Spin Systems.
Let us describe briefly, the quantum spin sys-tem and its dynamics. The observables at sites of the lattice Z d are elements of the algebra M ( C ) and the C ∗ -algebra of observables confined in the finite region X ⊂ Z d is A X := O x ∈ X M ( C ) . If L denotes the net of all finite subsets of Z d , directed by inclusion, the system { A X : X ∈ L} is in a natural way a net of C ∗ -algebras so that the algebra of all local observables given by A := [ X ∈L A X , is naturally normed and its norm completion is a C ∗ -algebra A (quasi-local observables).Interactions among particles in finite regions is represented by a family of self-adjoint elementsΦ := { Φ X : X ∈ L} ⊂ A X . Using the Pauli’s matrices σ xj ∈ M ( C ), j = 0 , , , , at the sites x ∈ Z d , in the isotropic, translation invariant, Heisenberg model , for example, in addition toan external potential represented by a one-body interaction of strength h ∈ R Φ( { x } ) := hσ x , particles interact only by a two-body potential so that Φ( X ) = 0 whenever | X | ≥ { x, y } ) := J ( x − y ) X i =1 σ xi σ yi x = y for a parameter λ > J : Z d → R describing the strength of the interactionbetween pairs of particles, under the assumption X x ∈ Z d e λ | x | | J ( x ) | < + ∞ . For any fixed Y ∈ L , the derivation A ∋ a i (cid:2) Φ Y , a (cid:3) ∈ A extends to a bounded derivation on A and generates a uniformly continuous group of auto-morphisms of A , representing the time evolution of the observables, interacting with thoseparticles confined in Y . To take into account simultaneously, the mutual influences amongparticles in different regions, one verifies that the superposition(9.1) D ( δ ) := A δ ( a ) := X Y ∈L i (cid:2) Φ Y , a (cid:3) , is a closable derivation on A whose closure is the generator of a strongly continuous group α Φ := { α Φ t : t ∈ R } of automorphisms of A .9.2. Markovian approach to equilibrium.
The above interactions provide the existence of( α Φ , β )-KMS-states ω at any inverse temperature β >
0. Let ( π ω , H ω , ξ ω ) be the GNS repre-sentations of the state ω , M the von Neumann algebra π ω ( A ) ′′ and ( M, L ( A, ω ) , L ( A, ω ) , J ω )the corresponding standard form. In the following we will use the smearing function f ( t ) :=1 / cosh(2 πt ). Theorem 9.1.
Suppose that ω is a ( α Φ , β ) -KMS-state at an inverse temperature satisfying (9.2) β < λ k Φ k λ where (9.3) k Φ k λ := sup x ∈ Z d X x ∈ X ∈L | X | | X | e λD ( X ) k Φ X k A X is finite under the exponential decay assumption on the strength J . Then the quadratic formsassociated to the self-adjoint elements a xj := π ω ( σ xj )(9.4) E x,j [ ξ ] = Z R k ( σ t − i/ ( a xj ) − j ( σ t − i/ ( a xj ))) ξ k f ( t ) dt are bounded completely Dirichlet forms and (9.5) E : L ( A, ω ) → [0 , + ∞ ] E [ ξ ] := X x ∈ Z d X j =0 E x,j [ ξ ] is a completely Dirichlet form on L ( A, ω ) . Concerning the proof, a first observation is that the E x,j are bounded Dirichlet form asuniformly convergent continuous superposition of elementary completely Dirichlet forms. Thequadratic form E is Markovian and closed as pointwise monotone limit of bounded completelyDirichlet forms. The only point that is left to be shown is the fact that it is densely defined,i.e. E is finite on a dense domain in L ( A, ω ). This is a consequence of the fact that underthe current hypotheses on the strength on the interaction, the dynamics has finite speedpropagation in the sense that, denoting by d ( x, X ) distance of the site x ∈ Z d from the region X ∈ L , we have(9.6) k [ α Φ t ( a ) , b ] k ≤ k a k · k b k · | X | · e − (cid:0) λd ( x,X )) − | t |k Φ k λ (cid:1) a ∈ A { x } , b ∈ A X , t ∈ R . Concerning the ergodic behaviour of the semigroups associated to the Dirichlet forms above,the following result shows how these properties are deeply connected to the other fundamentalproperties of the KMS-state.
Corollary 9.2. ([P2 Theorem 2.1]) Within the assumption of Theorem 3.3, the followingproperties are equivalent:i) ω is an extremal ( α Φ , β ) -KMS-state;ii) ω is a factor state in the sense that the von Neumann algebra M := π ω ( A ) ′′ is a factor.iii) the Markovian semigroup { T t : t ≥ } is ergodic in the sense that the subspace of L ( A, ω ) where it acts as the identity operator is reduced to the scalar multiples of the cyclic vector ξ ω ∈ L ( A, ω ) representing the KMS state ω . Extremality, i.e. the impossibility to decompose a KMS state as convex, nontrivial superpo-sition of other KMS states (see [BR2]), is the mathematical translation of the notion of purephase in Statistical Mechanics.Ergodicity of Markovian semigroups were considered by L. Gross [G1] to prove the uniquenessof the ground state of physical Hamiltonians in Quantum Field Theory. Later, S. Albeverioand R.Hoegh-Khron [AHK2] established a Frobenious type theory for positivity preservingmaps on von Neumann algebras with trace and in [Cip3] a Perron type theory was providedfor positivity preserving maps on the standard form of general von Neumann algebras.10.
Applications to Quantum Probability
As pointed out in Introduction, one of the major achievement of the theory of commuta-tive potential theory is the correspondence between regular Dirichlet forms and symmetricMarkov-Hunt processes on metrizable spaces. In noncommutative potential theory we donot dispose at moment of a complete theory but we have at least a clear connection be-tween Dirichlet forms of translation invariant, symmetric, Markovain semigroups and L´evy’sQuantum Stochastic Processes on Compact Quantum Groups.10.1.
Compact Quantum Groups d’apres S.L. Woronowicz. ([W]). In the following m A : A ⊗ alg A → A will denote the extension of the product operation of A .Let us recall that a compact quantum group G := ( A, ∆) is a unital C ∗ -algebra A =: C ( G )together with ai) coproduct ∆ : A → A ⊗ max A , a unital, ∗ -homomorphism which isii) coassociative (∆ ⊗ id A ) ◦ ∆ = ( id A ⊗ ∆) ◦ ∆ and satisfiesiii) cancellation rules : the closed linear span of (1 ⊗ A )∆( A ) and ( A ⊗ A ) is A ⊗ A . An example of the above structure arise from a compact group G by dualization of its struc-ture. In fact, setting A := C ( G ) we have A ⊗ max A = C ( G × G ) and a coproduct definedby (∆ f )( s, t ) := f ( st ) f ∈ C ( G ) , s, t ∈ G. A unitary co-representation of G is a unitary matrix U = [ u jk ] ∈ M n ( A ) such that∆ u jk = n X i =1 u ji ⊗ u ik j, k = 1 , · · · , n. Denote by b G the set of all equivalence classes of unitary co-representations of G . If a familyof inequivalent irreducible, unitary co-representations { U s : s ∈ G } of G exhausts all of b G ,then the algebra of polynomials , defined by the linear span of the coefficients of all unitaryco-representations Pol( G ) := linear span { u jk ∈ A : [ u jk ] ∈ b G } is a Hopf ∗ -algebra , dense in A , with counit ǫ and antipode S determined by ǫ ( u jk ) := δ jk , S ( u jk ) := u ∗ kj [ u jk ] ∈ b G and satisfying the rules( ǫ ⊗ id )∆( a ) = a, ( id ⊗ ǫ )∆( a ) = a, m A ( S ⊗ id )∆( a ) = ǫ ( a )1 A = m A ( id ⊗ S )∆( a ) . The C ∗ -algebra C ( G ) of a compact quantum group G is commutative if and only if it is ofthe form C ( G ) for some compact group G . In this case counit and antipode are defined by ǫ ( f ) := f ( e ) , S ( f )( s ) := f ( s − ) s ∈ G, where e ∈ G is the group unit.Combining the tensor product with the coproduct, one may introduce new operations thatin the case of compact group reduce to the well known classical ones.The convolution ξ ∗ ξ ′ ∈ A ∗ of functionals ξ, ξ ′ ∈ A ∗ is defined by ξ ∗ ξ ′ := ( ξ ⊗ ξ ′ ) ◦ ∆and the convolution ξ ∗ a ∈ A ∗ of a functional ξ ∈ A ∗ and an element a ∈ A is defined by ξ ∗ a = ( id ⊗ ξ )(∆ a ) a ∗ ξ := ( ξ ⊗ id )(∆ a ) . By a fundamental result of S.L. Woronowicz, on a compact quantum group G there existsa unique (Haar) state h ∈ A ∗ + which is both left and right translation invariant in the sensethat a ∗ h = h ∗ a = h ( a )1 A a ∈ A = C ( G ) . In the commutative case the Haar state reduces to the integral with respect to the Haarprobability measure. However, in general, the Haar state is not even a trace but it is a( σ, − σ t ∈ Aut( A ), t ∈ R , h ( ab ) = h ( σ − i ( b ) a ) a, b ∈ Pol( G ) . By a result of S.L. Woronowicz, the antipode S : Pol( G ) → C ( G ) is a densely defined, closableoperator on A and its closure ¯ S admits the polar decomposition¯ S = R ◦ τ i/ wherei) τ i/ generates a ∗ -automorphisms group τ := { τ t : t ∈ R } of the C ∗ -algebra A and ii) R is a linear, anti-multiplicative, norm preserving involution on A commuting with τ ,called unitary antipode .10.1.1. SU q (2) compact quantum group. The compact quantum group SU q (2) with q ∈ (0 , ∗ -algebra generated by the coefficients of a matrix U = (cid:20) α − qγ ∗ γ α ∗ (cid:21) subject to the relations ensuring unitarity: U U ∗ = U ∗ U = I . Then one may check that, interms of the generators α, γ , all the other relevant objects are determined byi) comultiplication: ∆( α ) := α ⊗ α + γ ⊗ γ , ∆( γ ) := γ ⊗ α + α ∗ ⊗ γ ii) counit: ǫ ( α ) := 1, ǫ ( γ ) := 0iii) antipode: S ( α ) := α ∗ , S ( γ ) := − qγ , S ( u j,k ) := ( − q ) ( j − k ) u − k,j for [ u jk ] ∈ b G iv) Haar state: h ( u jk ) := 0 for [ u jk ] ∈ b G v) automorphisms group: σ z ( u jk ) := q iz ( j + k ) u jk for [ u jk ] ∈ b G and z ∈ C vi) unitary antipode: R ( u jk ) := q k − j u ∗ jk for [ u jk ] ∈ b G .When q = 1 one recovers the classical compact group SU (2).10.1.2. Countable discrete groups as CQGs.
Let Γ be a countable discrete group and λ : Γ → B ( l (Γ)) its left regular representation λ s : l (Γ) → l (Γ) λ s ( δ t ) := δ st s, t ∈ Γ . The reduced C ∗ -algebra C ∗ r (Γ) ⊂ B ( l (Γ)) is the smallest C ∗ -algebra containing all the unitaryoperators λ s for s ∈ Γ. If instead of the regular representation one uses the direct sum of allcyclic unitary representation of Γ, the resulting algebra is called the universal C ∗ -algebra. Itis isomorphic to the regular one if and only if Γ is amenable.A compact quantum group structure on C ∗ r (Γ) is obtained extending to a ∗ -homomorphism∆ from C ∗ r (Γ) to C ∗ r (Γ) ⊗ C ∗ r (Γ) the map defined by ∆( λ s ) := λ s ⊗ λ s for s ∈ Γ. Thelinear span of the unitaries λ s for s ∈ Γ is a dense ∗ -Hopf agebra on which counit andantipode are defined as ǫ ( λ s ) = 1 and S ( λ s ) := λ s − for s ∈ Γ. The compact quantumgroup C ∗ r (Γ) is cocommutative in the sense that the comultiplication ∆ is invariant underthe flip of the left and right factors of C ∗ r (Γ) ⊗ C ∗ r (Γ). A theorem of Woronowicz ensuresthat any cocommutative compact quantum group C ( G ) is essentially the C ∗ -algebra of acountable discrete group in the sense that there exists a countable discrete group Γ and ∗ -homomorphisms C ∗ (Γ) → C ( G ) → C ∗ r (Γ). The CQG C ∗ r (Γ) is of Kac type and the Haarstate coincides with the trace determined by τ ( δ s ) = 0 for s = e and τ ( δ e ) = 1.10.2. L´evy processes on Compact Quantum Groups. ([CFK]). The L´evy processes oncompact groups are among the most investigated stochastic processes in classical probability.We briefly describe in this section a class of quantum stochastic processes, in the sense of[AFL] (see also [GS]), on compact quantum groups that generalize the classical L´evy processes.Let ( P, Φ) be a von Neumann algebra with a faithful, normal state, also called a noncom-mutative probability space .i) A random variable on G is a ∗ -algebra homomorphism j : Pol( G ) → P ii) the distribution of the random variable is the state φ j := Φ ◦ j on Pol( G )iii) the convolution j ∗ j of the random variable j , j : Pol( G ) → P is the random variable j ∗ j := m P ◦ ( j ⊗ j ) ◦ ∆ where m P is the product in P .A Quantum Stochastic Process ([AFL]) is a family of random variables { j s,t : 0 ≤ s ≤ t } satisfyingi) j tt = ǫ P for all 0 ≤ t ii) increment property : j rs ∗ j st = j rt for all 0 ≤ r ≤ s ≤ t iii) weak continuity : j tt → j ss in distribution as t → s decreasing. Definition 10.1. (Quantum L´evy Processes) A L´evy process on a CQG G is a quantumstochastic process on the Hopf-algebra Pol( G ) such that it hasi) independent increments in the sense that for disjoint intervals ( s k , t k ], k = 1 , · · · , n Φ( j s t ( a ) · · · j s n t ( a n )) = Φ( j s t ( a )) · · · Φ( j s n t ( a n ))ii) stationary increments in the sense that the distribution φ st = Φ ◦ j st depends only on t − s . Theorem 10.2.
Under a suitable probabilistic notion of equivalence of quantum stochasticprocesses, equivalence classes of L´evy processes { j s,t : 0 ≤ s ≤ t } on a compact quantumgroup G are in one-to-one correspondence with those Markovian semigroups { S t : 0 < t } onthe C ∗ -algebra C ( G ) which are translation invariant in the sense that ∆ ◦ S t = ( id ⊗ S t ) ◦ ∆ t > . To illustrate the main steps of the correspondence, notice first that the distributions of theprocess φ t := Φ ◦ j t form a continuous convolution semigroup on Pol( G ) φ = ǫ, φ s ∗ φ t = φ s + t , lim t → + φ t ( a ) = ǫ ( a ) a ∈ Pol( G )and that the generating functional of the process is then defined as G : D ( G ) → C G ( a ) := ddt φ t ( a ) (cid:12)(cid:12)(cid:12) t =0 on a dense domain D ( G ) ⊆ Pol( G ). From it one can reconstruct the distribution of theprocess as a convolution exponential φ t = exp ∗ ( tG ) := ǫ + ∞ X n =1 t n n ! G ∗ n t > , a semigroup on Pol( G ) by S t a = φ t ∗ a a ∈ Pol( G ) , t > L : Pol( G ) → Pol( G ) as L ( a ) := G ∗ a . Then one checks that thesemigroup extends to a strongly continuous, translation invariant Markovian semigropup onthe C ∗ -algebra C ( G ) and that its generator is the closure of L . Moreover, the distribution andthe generating functional can be written directly in terms of the semigroup and its generator φ t = ǫ ◦ S t , G = ǫ ◦ L. The KMS-symmetry of the semigroup of a L´evy process can checked using the generatingfunctional as follows
Theorem 10.3.
Let { S t : t > } be the Markovian semigroup of a L´evy process { j s,t : 0 ≤ s ≤ t } on compact quantum group G . The following properties are then equivalenti) the semigroup is ( σ h , − -KMS symmetricii) the generating functional is invariant under the action of the unitary antipode G = G ◦ R on the Hopf ∗ -algebra Pol( G ) . If the above conditions are verified then one can proceed to construct the Dirichlet formassociated to the L´evy process. The differential structure of these Dirichlet forms and thegenerating functional can be described in terms of the Sch¨urmann cocycle (see [CFK]) butwe do not pursue it here.Rather, we prefer to conclude this section with examples of Dirichlet forms on a class ofcompact quantum groups whose spectrum has been completely determined with applicationto the approximation properties treated in a previous section.10.2.1.
Free orthogonal Quantum Groups.
The universal C ∗ -algebra C u ( O + N ) of the free or-thgonal quantum group of Wang O + N , N ≥
2, is generated by a set of N self-adjoint elements { v jk : j, k = 1 , · · · , N } subject to the relations which ensure that the matrix [ v jk ] is unitary N X l =1 v lj v lk = δ jk = N X l =1 v jl v kl and where a coproduct is defined as ∆ v jk := P Nl =1 v lj ⊗ v lk . The Haar state is a trace whichis faithful on the Hopf algebra but not on C u ( O + N ) so that the L´evy semigroup is consideredon the reduced C ∗ -algebra C r ( O + N ), defined by the GNS representation of the Haar state. Theset of equivalence classes of irreducible, unitary co-representations is indexed by N . Denotingby { U s : s ∈ N } the Chebyshev polynomial on the interval [ − N, N ] defined recursively as U ( x ) = 1 , U ( x ) := x, U n ( x ) = xU n − ( x ) − U n − n ≥ , a generating functional is then defined by G ( u njk ) := δ jk U ′ n ( N ) U n ( N ) j, k = 1 , · · · , U n ( N ) , n ∈ N . It can be proved that the associated Dirichlet form has discrete spectrum whose eigenvec-tors are the coefficients u njk of the irreducible, unitary co-representations and such that thecorresponding eigenvalues and multiplicities are λ n := U ′ n ( N ) U n ( N ) , m n := ( U n ( N )) . By the results of a previous section, this implies that the von Neumann algebras L ∞ ( C ∗ r ( O + N ) , τ )generated by the GNS representation of the Haar trace states, all have the Haagerup Prop-erty. In particular, however, since for N = 2 one has λ n = n ( n +2)6 and m n = ( n + 1) , it resultsthat L ∞ ( C ∗ r ( O +2 ) , τ ) is amenable. The amenability of the free orthogonal quantum groupshave been proved for the first time by M. Brannan [Bra].10.2.2. Property (T) of locally compact quantum groups and boundedness of Dirichlet forms.
We conclude this exposition describing succinctly a recent result of A. Skalski and A. Viselter[SV] connecting the Property (T) of the von Neumann algebra of a quantum group to theboundedness of translation invariant Dirichlet forms. Their framework is more general thanthe one treated in this section as they consider the locally compact quantum groups G =( M, ∆ , ϕ H ), in the von Neumann algebra setting, of J. Kustermans and S. Vaes [KV].The main difference with respect to the S.L. Woronowicz theory of compact quantum groupsis that the Haar weight ϕ H (in general no more a state) is, together with the coproductoperation ∆, part of the structure of a locally compact quantum group G . This causes alack of certain common, dense, natural domain for generators, generating functionals and quadratic forms so that a subtler analysis is required.The von Neumann algebra M (resp. its standard space L ( M )) is often indicated as L ∞ ( G )(resp. L ( G )) or as L ∞ ( G , ϕ H ) (resp. L ( G , ϕ H )) to emphasize the reference to the chosenHaar weight.From the point of view of potential theory, the unboundedness of the Plancherel weightnecessitates of the extension of the theory of Dirichlet forms with respect to weights on vonNeumann algebras, developed by S. Goldstein and J.M. Lindsay in [GL3] (and amended in[SV Appendix]). We do not describe the details of this theory here but we just notice thatin case the Plancherel weight ϕ H is a trace we may use the theory illustrated in Section 4.The following result, obtained in [SV Theorem 4.6], characterizes the Property (T) of vonNeumann algebras of separable locally compact quantum groups (defined in [F] for discretequantum groups and for general locally compact ones in [DFSW]) in terms of a spectralproperty of the completely Dirichlet forms. Theorem 10.4.
Let G be a locally compact quantum group such that L ( G , ϕ H ) is separable.Then the following properties are equivalenti) the von Neumann algebra L ∞ ( G , ϕ H ) has the property (T)ii) any translation invariant completely Dirichlet form on L ( G , ϕ H ) is bounded. As in the compact case, the translation invariance of the Dirichlet form may be expressedas the invariance of the associated generating functional with respect to the unitary antipode.
Acknowledgements.
The author wishes to thanks the referee for her/his careful and pa-tient work.
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