KMS Dirichlet forms, coercivity and super-bounded Markovian semigroups on von Neumann algebras

We introduce a construction of Dirichlet forms on von Neumann algebras M associated to any eigenvalue of the Araki modular Hamiltonian of a faithful normal non tracial state, providing also conditions by which the associated Markovian semigroups are GNS symmetric. The structure of these Dirichlet forms is described in terms of spatial derivations. Coercivity bounds are proved and the spectral growth is derived. We introduce a regularizing property of positivity preserving semigroups (superboundedness) stronger than hypercontractivity, in terms of the symmetric embedding of M into its standard space L2(M) and the associated noncommutative Lp(M) spaces. We prove superboundedness for a special class of positivity preserving semigroups and that some of them are dominated by the Markovian semigroups associated to the Dirichlet forms introduced above, for type I factors M. These tools are applied to a general construction of the quantum Ornstein-Uhlembeck semigroups of the Canonical Commutation Relations CCR and some of their non-perturbative deformations.


Introduction and description of the results.
The structure of completely Dirichlet forms with respect to lower semicontinuous, faithful traces on von Neumann algebras is well understood in terms of closable derivations taking values in Hilbert bimodules (see [14] and the recent [36], [37]).However, for applications to Quantum Statistical Mechanics (see [4,5,6,7,10,11,22,23,25,27,28,29,30,38]) and Quantum Probability (see [12,26]) or to deal with general Compact Quantum Groups, is unavoidable to consider quadratic forms which are Markovian with respect to non tracial states or weights.Concerning the structure of Dirichlet forms of GNS-symmetric Markovian semigroups, one is invited to consult the recent [36], [37].In QSM, for example, the relevant states one wishes to consider are the KMS equilibria of time evolution automorphisms which are non tracial at finite temperature.In the CQGs situation, on the other hand, the Haar state is a trace only for the special subclass of CQGs of Kac type.In several most studied CQGs the Haar state is not a tracial state, as for examples for the special unitary CQGs SU q (N ).In this framework a detailed understanding has been found for the completely Dirichlet forms generating translation invariant completely Markovian semigroups of Levy quantum stochastic processes.The construction relies on the Schürmann cocycle associated to the generating functional of the process (see [13]).On the other hand, a general construction of completely Dirichlet forms on the standard form of a σ-finite von Neumann algebra with respect to a faithful, normal state in the sense of [8,9,16,17,18], has been introduced in [34,23,24] and by Y.M. Park and his school (see [28,29,30,4,5]) with applications to QSM of bosons and fermions system and their quasi-free states.In this approach the Dirichlet forms depend upon the explicitly knowledge of the modular automorphisms group of the state.
In this work we formulate a general and natural construction of a completely Dirichlet form, Markovian with respect to a fixed normal, faithful state ω 0 , associated to each non zero and not necessarily discrete eigenvalue of the Araki modular Hamiltonian ln ∆ 0 .Hence, by superposition, one has a malleable tool to construct completely Dirichlet forms and completely Markovian, modular symmetric, semigroups starting from the spectrum of the modular operator ∆ 0 or its associated Araki modular Hamiltonian ln ∆ 0 .Compared to Park's approach, this has the advantage to avoid the explicit use of the modular automorphism group.The present method generalizes the construction of bounded Dirichlet form of [8] Proposition 5.3 and that of unbounded Dirichlet forms of [8] Proposition 5.4, removing the assumption of self-adjointness and affiliation to the centralizer for the coefficients.The framework of the construction is that of Dirichlet forms and Markovian semigroups on standard forms (M, L 2 (M ), L 2 + (M ), J) of von Neumann algebras M as in [8] and related modular theory ( [1,2,6,35,33,39]).In particular, we associate in Section 2, a one-parameter family of unbounded, J-real, non negative, densely defined, closed quadratic forms (E λ Y , F λ Y ) on L 2 (M ) satisfying the first Beurling-Deny condition to each densely defined, closed operator (Y, D(Y )) affiliated to M , thus generating C 0 -continuous, contractive semigroups on L 2 (M ) which are positivity preserving (in the sense that they leave globally invariant the positive self-polar cone L 2 + (M )).Moreover, the quadratic form (E λ Y , F λ Y ) is Markovian with respect to the cyclic vector ξ 0 ∈ L 2 (M ) + representing ω 0 , in the strong sense that E λ Y [ξ 0 ] = 0, if and only if ξ 0 lies in the domain both of Y and its adjoint Y * and ξ := Y ξ 0 is an eigenvector of the modular operator ∆ 0 associated to the non zero eigenvalue λ > 0. This construction applies, in particular, to any eigenvector ξ of any non zero eigenvalue of ∆ 0 .Further, we investigate the fact that, by definitions, each Y , F λ Y ).By the general theory, using the symmetric embeddings of the von Neumann algebra M into the standard Hilbert space L 2 (M ) and the embedding of L 2 (M ) into the predual space M * = L 1 (M ), provided by the modular theory of the state ω, completely Markovian semigroups T t on L 2 (M ) extend to completely (Markovian) contractive semigroups on M and on L 1 (M ) (weak *continuous in the former case and strongly continuous in the latter one).In Section 4, we introduce an extra regularity property of positivity preserving semigroups called superboundedness as the boundedness of T t from L 2 (M ) to M for all t > t 0 and some t 0 ≥ 0. In case t 0 = 0 we call this property ultraboundedness.We prove that superboundedness holds true with respect to a finite temperature Gibbs state ω(•) = Tr (•e −β0H0 )/Tr(e −β0H0 ) on a type I ∞ factor M , for the semigroup generated by the generalized sum H 0 +JH 0 J and that the property is stable with respect to domination of positivity preserving semigroups.In Section 5 we apply the framework above to investigate a class of Dirichlet forms associated on a type I ∞ factor which are Markovian with respect to a Gibbs state of the Number Operator of a representation of the CCR algebra.The construction fully generalizes that of Quantum Ornstein-Uhlenbeck semigroups introduced in [12].In particular we prove the subexponential spectral growth rate of the generator and the domination of the Markovian semigroup with respect to the semigroup generated by H 0 +JH 0 J (this special class of semigroups is discussed in Appendix 7.1).In Section 6 we apply the tools developed in the previous sections to construct Dirichlet forms associated with dynamics generated by deformations of the Number Operator.
In Appendix we represent the generators of a class of positivity preserving semigroups as generalized sums and we clarify superboundedness for abelian von Neumann algebras.

Dirichlet forms and derivations on von Neumann algebras standard forms
Let (M, L 2 (M ), L 2 + (M ), J) be a standard form of a σ-finite von Neumann algebra (for this subject and the related modular theory we refer to [6,7,35,39]).Let ω 0 be the faithful normal state on M represented by the cyclic vector The anti-linear, densely defined operator on L 2 (M ) defined on the left Hilbert algebra by where the anti-unitary part J is called the modular conjugation and ∆ 0 := S * 0 S 0 is a densely defined, self-adjoint, positive operator on L 2 (M ), called the modular operator of ω 0 , defining the modular automorphism group of M by σ ω0 t (x) := ∆ it 0 x∆ −it 0 for x ∈ M and t ∈ R. On the w * -dense, involutive, sub-algebra of its analytic elements M 0 ⊆ M , the modular group can be extended to any t ∈ C. For any x, y ∈ M 0 and z, w ∈ C, this extension satisfies ).We will make use of the symmetric embedding of M into its standard Hilbert space L 2 (M ): 0 xξ 0 .Among its properties we recall that it is weak * -continuous, injective with dense range and positivity preserving in the sense that i 0 (x) ∈ L 2 (M ) + if and only if x ∈ M + .Also it maps the closed and convex set of all x ∈ M + such that 0 ≤ x ≤ 1 onto the closed and convex set of all ξ ∈ L 2 + (M ) such that 0 ≤ ξ ≤ ξ 0 .The projection of a J-real vector ξ = Jξ ∈ L 2 (M ) onto the closed, convex set ξ 0 − L 2 + (M ) wil be denoted by ξ ∧ ξ 0 .A Dirichlet form [8] Definition 4.8 with respect to (M, ω 0 ) is a lower bounded and lower semicontinuous quadratic form (E, F ) is said to be a completely Dirichlet form if its ampliation on the algebra (M ⊗ M n (C), ω 0 ⊗ tr n ) defined by is a Dirichlet form for all n ≥ 1 (tr n denotes the tracial state on the matrix algebra M n (C)).
As a result of the general theory, Dirichlet forms are automatically nonnegative and Markovian semigroups are automatically contractive see [8] Proposition 4.10 and Theorem 4.11.Dirichlet forms (E, F ) are in one-to-one correspondence with Markovian semigroups {T t : t ≥ 0}: the self-adjoint, positive operator (H, D(H)) associated to (E, F ) by for all ξ ∈ F , being the semigroup generator T t = e −tH , t ≥ 0. C 0 -continuous, self-adjoint, positivity preserving semigroups are in one-toone correspondence with nonnegative, densely defined, real, lower semicontinuous quadratic forms satisfying the following first Beurling-Deny condition (weaker than Markovianity) equivalently stated (see [8] Proposition 4.5 and Theorem 4.7]) as On the other hand, the first Beurling-Deny condition and the conservativeness condition together imply the Markovianity of closed forms (E, F ) (see [8] Lemma 2.9 and Theorem 4.11).

Dirichlet forms associated to eigenvalues of the modular operators
The forthcoming construction of Dirichlet forms is based on the following well known fact (see [6] Proposition 2.5.9, [33,35] page 19; see also [2] where von Neumann algebras with states having the logarithmic of the modular operators with spectrum consisting only of isolated eigenvalues are characterized).
For any operator (Y, D(Y )) affiliated to M , the operator j(Y iii) Among the operators (Y, D(Y )) with the properties i) and ii) above, there exists a minimal one (Y 0 , D(Y 0 )) obtained as the closure of the closable operator (Y 0 , D(Y 0 )) defined by Proof.The operator (Y 0 , D(Y 0 )) is affiliated to M because the action of any w ′ ∈ M ′ leaves globally invariant the domain M ′ ξ 0 and w ′ Y 0 (y ) is closable because it is in duality with the densely defined operator Z 0 : M ′ ξ 0 → L 2 (M ) given by Z 0 (z ′ ξ 0 ) := z ′ S 0 ξ in the sense that Clearly by definition Y 0 ξ 0 = ξ and the calculation above implies ) is a closed operator affiliated to M with properties i) and ii) above, then, as This representation will be applied below to eigenvectors ξ (if any) of the modular operator.Lemma 2.2.Let (Y, D(Y )) be a densely defined, closed operator affiliated to M and µ, ν ≥ 0. Then defining ) is a densely defined, closable operator on L 2 (M ).
Proof.Since J 2 = I, we have D(j(Y * )) = JD(Y * ) so that d µ,ν Y is well defined on D(d µ,ν Y ).By hypotheses, j(Y * ) is densely defined, closed and affiliated to the commutant von Neumann algebra M ′ .Hence Y and j(Y * ) strongly commute and the contraction semigroup e −t|Y | •e −t|j(Y * )| = e −t|j(Y * )| •e −t|Y | with parameter t ≥ 0 strongly converges to the identity operator on L 2 (M ) ) is a bounded operator for any t > 0, we have that D(d µ,ν Y ) is dense in L 2 (M ).To prove the statement concerning closability, observe that reasoning as above with Y * in place of Y and Proof.Consider first the case where Y is bounded, and let s ′ ± ∈ M ′ be the supports in M ′ of the positive and negative parts ξ ± of a J-real ξ ∈ L 2 (M ). To By the Dominated Convergence Theorem we have ) is positivity preserving.
) and satisfies the first Beurling-Deny condition satisfies the first Beurling-Deny condition too and generates a contractive, positivity preserving semigroup {T t : t ≥ 0}.Moreover, completely Dirichlet form with respect to (M, ω 0 ) and the associated completely Markovian semigroup is conservative, in the sense that if and only if Y ξ 0 ∈ L 2 (M ) is an eigenvector of the modular operator corresponding to the eigenvalue µ/ν Y * ξ for all ξ ∈ FY and, exchanging the role of Y and Y * , we have which proves that the quadratic form ( Ẽµ,ν Y , FY ) is J-real.Consider now a J-real vector ξ ∈ FY , its polar decomposition ξ = ξ + − ξ − with respect to the self-polar cone L 2 + (M ) and recall that, by definition, |ξ| := ξ + +ξ − .By the previous lemma, |ξ| ∈ FY so that ξ To establish the same condition for the closure (E µ,ν Y , F µ,ν Y ), we adapt the proof of [8] Proposition 5.1 (according the suggestions which there precede it).On one hand, since J is an isometry for the graph norm of ( Ẽµ,ν Y , FY ) and FY is a core for (E µ,ν Y , F µ,ν Y ),then J is an isometry for the latter form too and the closure form is J-real.On the other hand, let ξ = Jξ ∈ F µ,ν Y be a fixed J-real vector and let ξ n ∈ FY be a sequence converging to it in the graph norm of (E µ,ν Y , F µ,ν Y ).Since the Hilbert projection η → η + of the J-real part of L 2 (M ) onto the closed, convex cone L 2 + (M ) is norm continuous and |η| = 2η + − η, it follows that the modulus map η → |η| is norm continuous too.Then, since the form The first Beurling-Deny condition is thus verified and, by [8] Proposition 4.10, it follows that the semigroup {T t : t ≥ 0} has the desired properties.
Concerning the conservativeness property, notice that By [8] Proposition 4.10, given conservativness, the first Beurling-Deny property and Markovianity are equivalent for quadratic forms as well as the positivity preserving property and the Markovianity are equivalent for the associated semigroups.Concerning the complete Markovianity of the Dirichlet form, we notice that for any n ≥ 1, the ampliation ( of the state ω 0 on M , corresponding to the eigenvalue µ/ν, then, denoting by an eigenvalue of the modular operator of the state ω 0 ⊗ tr n on M ⊗ M n (C), corresponding to the same eigenvalue µ/ν.Applying the results obtained above to the form (E λ Y ) n in place of E λ Y , we get its Markovianity for any n ≥ 1 and complete Markovianity of the associated semigroup.

Y
. Since now on, we will adopt the simplified notation Remark.To any eigenvector ξ ∈ D(S 0 ) of ∆ x, y ∈ M, t ≥ 0 and, in particular, it commutes with )) for the eigenvalue λ 2 and, by Theorem 2.5, (E λ Yt , F λ Yt ) is a well defined Dirichlet form.The displayed identity follows from the identities valid, for any t ∈ R, on FYt = FY and the fact that this space is a form core for η ) is invariant under the unitary group {∆ it 0 : t ∈ R} so that the Markovian semigroup it generates commutes with {∆ it 0 : t ∈ R} and it is GNS symmetric by [8] Theorem 6.6; v) follows from iv) as, by definition, M ′ ξ 0 is a core for (Y 0 , D(Y 0 )).

Representation of Dirichlet forms as square of commutators
In this section we show how to represent the Dirichlet forms on L 2 (M ) constructed above, in terms of generalized commutators, i.e. unbounded spatial derivations on M .
We recall that (S 0 , D(S 0 )) is an unbounded conjugation, i.e. anti-linear and idempotent on its domain.Thus S 2 0 is the identity operator on D(S 0 ) or, more explicitly, that ξ ∈ D(S 0 ) implies S 0 ξ ∈ D(S 0 ) and S 0 (S 0 ξ) = ξ.In other terms, the image of S 0 coincides with its domain and S 0 = S −1 0 holds true as an identity between densely defined, closed operators.In terms of the polar decomposition S 0 = J∆ 1/2 0 we have J∆ J as an identity between densely defined, closed operators.This means, in particular, that the modular conjugation exchanges domains as follows JD(∆ ).More in general, one has the intertwining relation f (∆ −1 0 ) = Jf (∆ 0 )J between closed operators valid for any Borel measurable function f : [0, +∞) → C (see Introduction to Chapter 10 in [39]).The relation, which is equivalent to JD( )), will be mostly used for power functions f .Among its consequences, we will make use of the following: a) for any α ∈ R, the closed operator J∆ α 0 is an unbounded conjugation on its domain is an identity between densely defined, closed operators: in fact, D(∆ e) Let M 0 ⊆ M be the involutive w * -dense sub-algebra of analytic vectors of the group σ ω0 .For any y ∈ M 0 , the operator ∆ 1/4 0 y∆ −1/4 0 on L 2 (M ) is densely defined on i 0 (M 0 ) and closable.Its closure is a bounded operator belonging to M , which coincides with the analytic extension of the map −i/4 (y)i 0 (x) = i 0 (yx) for all x ∈ M 0 ; f) by Proposition in Section 9.24 in [39], for any y ∈ M 0 and any α ∈ C one has the important identity ), the case α = 1/2 implies that D(S 0 yS 0 ) = D(S 0 ) and the boundedness of the operator S 0 yS 0 on D(S 0 ); g) the involutive sub-algebra M ′ 0 := JM 0 J ⊂ M ′ coincides with the set of analytic vectors of the modular group of the commutant M ′ associated to the state determined by ξ 0 ∈ L 2 (M ).The left Hilbert sub-algebra M 0 ξ 0 ⊂ M ξ 0 ⊂ L 2 (M ) is dense in L 2 (M ) and it coincides with the symmetric embedding of the algebra of analytic elements as it results from the identity i 0 (y) = σ ω0 −i/4 (y)ξ 0 valid for all y ∈ M 0 .Also, Lemma 2.7.If η ∈ D(S 0 ), the densely defined operator (L η , D(L η )) given by The densely defined operator (R η , D(R η )) given by The relation between the operators L η and R η follows from the identities i 0 (y 0 yξ 0 = Ji 0 (y) for all y ∈ M 0 and the fact that J is idempotent and it leaves Lemma 2.8.Let ξ ∈ D(S 0 ) and fix, by Lemma 2.1, a densely defined, closed operator (X, D(X)) affiliated to M such that Then the following properties hold true: ii) the images of M 0 ξ 0 under (X, D(X)) and (X * , D(X * )) are contained in D(S 0 ) Consider the densely defined operators on L 2 (M ) given by is well defined on i 0 (M 0 ) and there it coincides with L ξ ; vii) the operator J∆ ii) Since M 0 ξ 0 is a core for (S 0 , D(S 0 )), there exists a sequence x n ∈ M 0 such that x n ξ 0 − Xξ 0 → 0 and x * n ξ 0 − X * ξ 0 → 0. As mentioned at item f) of the introduction of the present section, since y ∈ M 0 , the operator S 0 y * S 0 is bounded on D(S 0 ) and then on M 0 ξ 0 ⊂ D(S 0 ).Thus x n yξ 0 ∈ D(S 0 ) is a Cauchy sequence in L 2 (M ) as Analogously, S 0 (x n yξ 0 ) = y * x * n ξ 0 ∈ L 2 (M ) is a Cauchy sequence too as Hence x n yξ 0 ∈ D(S 0 ) is a Cauchy sequence in the graph norm of the closed operator (S 0 , D(S 0 )) and the image of η := lim n x n yξ 0 ∈ D(S 0 ) is given by By the density of M ′ ξ 0 in L 2 (M ), it follows that Xyξ 0 −x n yξ 0 ≤ S 0 y * S 0 • Xξ 0 −x n ξ 0 → 0 as n → ∞ and we have Xyξ 0 = η ∈ D(S 0 ) and S 0 (Xyξ 0 ) = S 0 (η) = y * X * ξ 0 for any y ∈ M 0 .As S 2 0 is the identity operator on D(S 0 ), from S 0 (Xξ 0 ) = X * ξ 0 it follows that X * ξ 0 ∈ D(S 0 ) and S 0 (X * ξ 0 ) = Xξ 0 .Thus (X * , D(X * )) satisfies the same hypotheses as (X, D(X)) and the statements involving (X * , D(X * )) can be deduced from those involving (X, D(X)) proved above, by substitution and the fact that the sub-algebra M 0 is involutive.
) so that R ξ is well defined too.
As first step to prove iii), we show that D(∆ ) and implies Since i 0 is positivity preserving, setting c := σ ω0 −i/4 (y) 2 , we thus obtain the bound 2 → 0 and, by analogous identities, the selfpolarity of L 2 + (M ) and the bound above, we get ∆ Coming back to the proof of iii), notice that, by the identity J∆

J, we have
0 Xξ 0 and we conclude the proof of iii) by To prove iv), notice first that, since S 2 0 is the identity operator on D(S 0 ) and Xξ 0 ∈ D(S 0 ), we have To prove v), i.e. that L ξ is affiliated with M , let us start to notice that for z ′ ∈ M ′ 0 and y ∈ M 0 , setting z := J∆ Ji 0 (y).By i) and ii), y * ξ 0 ∈ D(X * ), X * y * ξ 0 ∈ D(S 0 ) and yXξ 0 = S 0 (X * y * ξ 0 ) ∈ D(S 0 ) ⊂ D(∆ J is densely defined on i 0 (M 0 ) and there it coincides with R ξ .
To prove the first identity in viii), notice that, by the Spectral Theorem, ξ = Xξ 0 is an eigenvector of ∆ 1/4 0 with eigenvalue λ > 0: ∆ By the density of M ′ 0 ξ 0 in L 2 (M ) and for all z ′ ∈ M ′ 0 we then have To prove the second identity in viii), we first need to show that the adjoint of the densely defined operator (∆ 1/4 0 X∆ −1/4 0 , i 0 (M 0 )) (which is closable by ii) and vi)) is an extension of the well defined and densely defined operator (∆ −i/4 (x * )ξ 0 .Since JX * J is affiliated to M ′ , σ ω0 −i/4 (x * ) ∈ M and by ii) σ ω0 −1/4 (x * )ξ 0 ∈ D(JX * J), we have The hypothesis that Xξ 0 is an eigenvalue of ∆ . For all y ∈ M 0 we may then compute ) * xξ 0 .Since by vi) and the first identity proved above we have To finalize the proof of the second identity in viii), rewrite the eigenvalue equation satisfied by ξ = Xξ 0 as JX * Jξ 0 = ∆ 1/2 0 Xξ 0 = λ 2 Xξ 0 so that, for all y ∈ M 0 , we have Lemma 2.9.Let ξ ∈ D(S 0 ) be eigenvector of ∆ 1/2 0 corresponding to the eigenvalue λ 2 > 0 and fix a densely defined, closed operator (X, D(X)) affiliated to M such that Proof.On one hand we have S 0 ξ = J∆ 1/2 0 ξ = λ 2 • Jξ.On the other hand, since JD(∆

Combining the results obtained, we have
Corollary 2.10.Let ξ ∈ D(S 0 ) be an eigenvector of ∆ 1/2 0 corresponding to the eigenvalue λ 2 > 0 and fix a densely defined, closed operator (X, D(X)) affiliated to M such that Then for all y ∈ M 0 we have and The commutator [X, y] := Xy − yX is in general only densely defined if X is affiliated to M but, within the hypotheses assumed at the beginning of this section, the vector ξ 0 belongs to the domain of [X, y] and its image [X, y]ξ 0 belongs to D(∆ 1/4 0 ).This may justify the notation In the following we will use the notation j(X * ) := JX * J.
Next result shows that the symmetric embedding i 0 intertwines the unbounded spatial derivations δ X , δ X * on M with the unbounded bimodule derivations corresponding to the eigenvalue λ 2 > 0 and fix a densely defined, closed operator (X, D(X)) affiliated to M such that Otherwise stated, setting δ X (y) := i[X, y] for any y ∈ M 0 , on the * -algebra M 0 we have Proof.For y ∈ M 0 we have σ ω0 −i/4 (y) ∈ M 0 , Jσ ω0 −i/4 (y)J ∈ M ′ 0 and Since X * is affiliated with M and ξ 0 ∈ D(X * ), we have (Jσ ω0 −i/4 (y)J)ξ 0 ∈ D(X * ) and ).The proof of the second identity is similar.Theorem 2.12.Let ξ ∈ D(S 0 ) be an eigenvector of ∆ 1/2 0 corresponding to the eigenvalue λ 2 > 0 and (X, D(X)) a densely defined, closed operator affiliated to M such that Then the completely Dirichlet form (E λ X , F λ X ) constructed above may be represented as Remark 2.13.These results prove a fortiori that and under the stated assumptions, the form extends to a completely Dirichlet form on L 2 (M ) with respect to the cyclic vector ξ 0 ∈ L 2 + (M ).If ξ 0 would be the vector representing a finite, normal, faithful trace state ω 0 , this result would follow from the general theory relating completely Dirichlet forms and closable bimodule derivations on von Neumann algebras with trace (see [14]).

Coercivity of Dirichlet forms
In this section we still keep the assumption that ξ ∈ D(S 0 ) is an eigenvector of ∆ 1/2 0 corresponding to the eigenvalue λ 2 > 0 (we still assume λ > 0) and (X, D(X)) a densely defined, closed operator affiliated to M such that We prove below natural lower bounds on the Dirichlet form (E λ X , F λ X ) constructed in Section 2, which lead to coercivity.Recall that (E λ X , F λ X ) is defined as the closure of the densely defined, J-real, closable quadratic form Ẽλ where Obviously FX is a form core for (E λ X , F λ X ) and on it E λ X and Ẽλ X coincide.We start showing an alternative representation of the Dirichlet form.
Theorem 3.1.The following representation holds true for the quadratic form Proof.In the following, we repeatedly use the fact that if N ⊆ B(h) is a von Neumann algebra acting on a Hilbert space h and (A, D(A)), (B, D(B)) are densely defined, closed operator on h affiliated to N and N ′ , respectively, then This identity follows directly if In general we may approximate B weakly by We start the proof of the result setting )) and using the splittings X * , for any η ∈ FX to have the representation the sum of the first two addends in (3.2) equals Since also the sum of the third and fourth addends in (3.2) equals the sum of the fifth and sixth addends in (3.2) equals and, analogously, the sum of the seventh and eighth addends in (3.2) equals By substitution of (3.6), (3.5) and (3.4) in (3.2) we obtain and then, by (3.3), we finally obtain (3.1) for any η ∈ FX Corollary 3.2.(Lower bound) The following lower bounds hold true for any ε, δ > 0 and any η ∈ FX In particular, for ε = δ = 1 and any η ∈ FX we have The result follows from (3.1) and the identities, valid for ε, δ > 0, We address now the problem to find conditions on (X, D(X)) sufficient to guarantee that the lower bounds above are coercive for our Dirichlet form.By this we mean bounds in which the Dirichlet form dominates a quadratic form with a certain degree of discreteness of the spectrum such as existence and finite degeneracy of a ground state, spectral gaps or emptiness of essential spectrum.The conditions will be formulated in terms of relative smallness of the quadratic form of the self-commutator [X, X * ] with respect to the quadratic form of X * X and they will be exploited in Section 5 when M is a type I ∞ factor.Let us denote by (t X , D(t X )) and (t X * , D(t * X )), the densely defined, positive, closed quadratic forms defined as whose associated positive, self-adjoint operators are (X * X, D(X * X)) and (XX * , D(XX * )).Consider also the quadratic form (q λ X , D(q λ X )) given by qλ . By the densely defined quadratic form (q 0 , D(q 0 )) defined as • q 0 and regard qλ X as a perturbation of a multiple of t X or t X * by a multiple of q 0 .Notice that q 0 is the form of the self-commutator [X, X * ] = XX * − X * X, at least on D(X * X) ∩ D(XX * ).Using the quadratic form ( Qλ X , FX ) given by Qλ the lower bound (3.8) can be written as Although Qλ X is densely defined, since i 0 (M 0 ) = M 0 ξ 0 ⊂ FX by Lemma 2.8 ii), it is not necessarily lower bounded, closable or a proper functional.For sake of clarity, we recall some definition we will use concerning lower bounded quadratic forms (A, D(A)), (B, D(B)) and their associated selfadjoint operators (A, D(A)), (B, D(B)) on a Hilbert space h (see [19]

Superboundedness of a class of semigroups on type I von Neumann algebras
In this section we introduce a further continuity property, called superboundedness, for positivity preserving semigroups on standard forms of σ-finite von Neumann algebras, showing that the property is owned by a class of semigroups on type I ∞ factors.Also we show how this property persists under domination of positivity preserving semigroups.
As usual, i 0 : M → L 2 (M ) denotes the symmetric embedding of a σ-finite von Neumann algebra M endowed with a faithful normal state ω 0 ∈ M * + represented by ξ 0 ∈ L 2 + (M ).Definition 4.1.(Excessive vectors and superboundedness) i) The vector ξ 0 ∈ L 2 + (M ) is (γ 0 , t 0 )-excessive or excessive, for some γ 0 , t 0 ≥ 0, with respect to a positivity preserving semigroup {T t : t ≥ 0} on L 2 (M ) if the maps e −γ0t T t are Markovian w.r.t.ξ 0 for any t > t 0 .Markovian semigroups are just those for which ξ 0 is (0, 0)-excessive; ii) a positivity preserving semigroup If we endow the subspace i 0 (M ) ⊆ L 2 (M ) by the norm of the von Neumann algebra, i.e. i 0 (x) M := x M for x ∈ M , then superboundedness implies the boundedness of T t as a map from (L 2 (M ), • 2 ) to (i 0 (M ), • M ) for all t > t 0 .In fact, by the norm continuity of the symmetric embedding i 0 : M → L 2 (M ), the norm • M is stronger than the Hilbert norm • 2 so that the continuous maps T t : L 2 (M ) → L 2 (M ) are closed when considered from the Hilbert space L 2 (M ) to the Banach space (i 0 (M ), • M ) and, by the Closed Graph Theorem, they result to be bounded (notice that this involves only condition b) in Definition 4.1).We shall refer to part b) of superboundedness writing T t L 2 (M)→M < +∞ for all t > t 0 and to part b) of supercontractivity writing T t L 2 (M)→M ≤ 1 for all t > t 0 .By the Markovianity of e −γ0t T t required in i), bounded, positivity preserving maps S t : M → M satisfying the relations i 0 (S t (x)) = T t (i 0 (x)) for x ∈ M are well defined and one has, for suitable scalars b t ≥ 0, Consider the noncommutative spaces L p (M, ω 0 ) for p ∈ [2, +∞] defined by the symmetric embedding i 0 : M → L 2 (M ) (see [21]).By complex interpolation it follows that a superbounded semigroup is hypercontractive too in the sense that there exists T 0 ≥ 0 such that T t is bounded from L 2 (M ) to L 4 (M, ω 0 ) for t > T 0 .
The following observation will be useful later on.

A class of superbounded Markovian semigroups on a type I ∞ factor
Let h be a Hilbert space and consider the type I factor M := B(h).Its (Hilbert-Schmidt) standard representation acts, by left composition, on the space L 2 (M ) = L 2 (h) of Hilbert-Schmidt operators on h, where the standard cone L 2 + (M ) = L 2 + (h) is that of operators in L 2 (h).The modular involution is given by the operator adjoint: Jξ := ξ * for ξ ∈ L 2 (h) and the right representation of B(h) on L 2 (h) is given by right composition.
Let H 0 be a lower bounded, self-adjoint operator affiliated to B(h) (i.e.any self-adjoint, lower bounded operator on h) and consider the strongly continuous semigroup on L 2 (h) given by Its self-adjoint generator G 0 on L 2 (h), defined by G 0 (ξ) := lim t→0 t −1 (ξ−T t ξ) on the subspace D(G 0 ) ⊂ L 2 (h) for whose vectors the limit exists, coincides with the generalized sum H 0 +JH 0 J (see [19]) of the closed operators H 0 and JH 0 J, affiliated to the commuting von Neumann algebras given by the left and right representations of B(h) on L 2 (h) (see Lemma 7.1 in Appendix).
The operator H 0 , resp.JH 0 J, is considered here as acting on a suitable dense subspace of the Hilbert-Schmidt space L 2 (h) by left, resp.right, composition.For example, G 0 (ξ) = H 0 • ξ + ξ • H 0 ∈ L 2 (h) for those ξ ∈ L 2 (h) such that the operators H 0 • ξ and ξ • H 0 are densely defined, closable and bounded on their domains and their closures are Hilbert-Schmidt operators.To ease notation, the operators H 0 • ξ, ξ • H 0 will be represented by the juxtaposition H 0 ξ, ξH 0 of the symbols of the operators H 0 and ξ so that, the formula above appears G 0 (ξ) = H 0 ξ + ξH 0 .For further details on Hilbert-Schmidt standard form we refer to [12] Section 2.
Lemma 4.3.If H 0 has discrete spectrum Sp(H 0 ) := {λ j : j ∈ N}1 with the increasing eigenvalues written with repetitions according to the their multiplicity, then i) G 0 has discrete spectrum too given by Sp(G 0 ) : Proof.Let H 0 = ∞ k=0 λ k P k be the spectral decomposition of H 0 as an operator acting on h.Then the spectral decomposition of G 0 is given by since {P j JP k J : j, k ≥ 0} is a complete family of mutually orthogonal projections acting on the standard Hilbert space L 2 (h) such that Thus G 0 has the discrete spectrum indicated in the statement and since λ j + λ k ≤ λ implies both λ j + λ 0 ≤ λ and λ 0 + λ k ≤ λ, the bound n G0 (λ) ≤ n N (λ − λ 0 ) 2 holds true for λ ∈ R.
Suppose now the lower bounded, self-adjoint operator H 0 on h to have a discrete spectrum Sp(H 0 ) := {λ j : j ∈ N} such that, for some β > 0, Tr(e −βH0 ) = ∞ k=0 e −βλ k < +∞, so that the Gibbs state on B(h) with density matrix ρ β := e −βH0 /Tr(e −βH0 ) is well defined and its representative positive vector is given by ξ Recall that in this case the symmetric embedding i 0 : B(h) → L 2 (h) is given by i 0 (x) = ρ In particular, if λ 0 ≥ 0, the semigroup is Markovian and supercontractive.

General quantum Ornstein-Uhlenbeck semigroups
In this section we apply the above framework to construct a family of Dirichlet forms and Markovian semigroups, a special case of which is the quantum Ornstein-Uhlenbeck semigroup studied in [12].While in [12] we computed explicitly the spectrum of the generator and proved the Feller property with respect to the algebra of compact operators, here we prove, for each semigroups we construct, subexponential spectral growth rate and domination with respect to positivity preserving semigroups belonging to a natural related class (see Appendix 7.1).
On the Hilbert space h := l 2 (N), consider the C * -algebra of compact operators K(h).The Number Operator (N, D(N )), defined by the natural basis e := {e k ∈ l 2 (N) : k ∈ N} as For any β > 0 there exists a unique (α, β)-KMS state ω β , satisfying the KMS condition ω β (Aα iβ (B)) = ω β (BA) for α-analytic elements A, B, given by, in terms of the density matrix, The extension of the automorphisms group α to a C * 0 -continuous group on B(h) is given by the same formula above on K(h).In the Hilbert-Schmidt standard form of M := B(h) described in Section 4.1, the cyclic and separating vector representing ω β is given by + (h).The action of the Hilbert algebra unbounded conjugation operator S 0 on L 2 (h), characterized as S 0 (xξ 0 ) := x * ξ 0 for x ∈ B(h), can be identified on a suitable domain D(S 0 ) ⊂ L 2 (h) with and its polar decomposition S 0 = J∆ iii) the densely defined, closed operator (X, D(X)) on h, given by where A is the annihilation operator defined in Section 5, satisfies the relations ) is an eigenvector of ln ∆ 0 with eigenvalue ν: the assumption on ε and the Spectral Theorem; iii) by the CCR we have as identities among closed operators on their common domain D(N 3/2 ).By induction on the domain D(N ℓ ) so that, by (6.2), one gets the first relation (6.3) Since, by (6.4), p(N )A = Ap(N − I), by induction one obtains the second relation ( 3) follows by difference; iv) follows from (6.3) and the fact that , by the Spectral Theorem, the spectral projection P of ln ∆ 0 , corresponding to {ν}, can be represented as ) by (6.1), by (6.4) we have Hence, P (A ℓ ξ 0 ) = R dtf (t)e itk(N ) (A ℓ (ξ 0 )) = ( f (k(N ))A ℓ )(ξ 0 ) = X(ξ 0 ) =: ξ does not vanish and it is an eigenvector of ln ∆ 0 corresponding to the eigenvalue ν.
Example. 1) If g(t) = t for any t ∈ R, B = N, p(N ) = I, X = A ℓ and we reproduce the "unperturbed" case treated in Theorem 5.2.
Remark 6.2.The canonical commutation relations CCR arise in the spectral analysis of the quantum harmonic oscillator, which can be considered the canonical quantization of the classical harmonic oscillator whose phase space is the plane R 2 .D. Shale and W. F. Stinespring constructed in [32] a quantum system which can be regarded as the quantization of a harmonic oscillator whose phase space is the hyperbolic plane H 2 with a fixed negative constant curvature k < 0. It can be also considered as a quantum harmonic oscillator with self-interaction, the coupling constant being proportional to the curvature.In their work the authors found that the dynamics is generated by an Hamiltonian H = ωN proportional to the Number Operator and that annihilation and creation operators are replaced by operators X and X * satisfying a deformed CCR In reference to Section 5, e −tH λ 2 X 2 , compared with the quantum Ornstein-Uhlenbeck semigroup e −tH λ 1 X 1 (see [12]), could be called quantum Ornstein-Uhlenbeck hyperbolic semigroup.If (q A , D(q A )) is the lower bounded, closed quadratic form of (A, D(A)), then the lower bounded, closed quadratic form of (j(A)), JD(A)) is given by JD(q A ) ∋ η → q A [Jη] and the quadratic form (t A , D(q A ) ∩ JD(q A )) given by t A [η] := q A [η] + q A [Jη] is lower bounded and closed as a sum of forms sharing these same properties.

Appendix
Lemma 7.1.The lower bounded, closed, quadratic form of the C 0 -continuous, self-adjoint semigroup {T A t : t ≥ 0} is given by (t A , D(q A ) ∩ JD(q A )) and the associated self-adjoint generator, i.e. the generalized sum A +j(A) (see [19]), is given by the closure A + j(A) T A t = e −tA j(e −tA ) = e −t(A +j(A)) = e −tA+j(A) t ≥ 0.
A C 0 -continuous semigroup T t : L 2 (X, m) → L 2 (X, m) is Markovian with respect to m U (in the sense we are discussing in this work, i.e. the one introduced in [8]), if Such a semigroup induces a semigroup on the abelian von Neumann algebra L ∞ (X, m) by S t : L ∞ (X, m) → L ∞ (X, m) i 0 (S t u) = T t (i 0 (u)) u ∈ L ∞ (X, m), which is Markovian in the usual sense The definition of superboundedness considered above on von Neumann algebras, in the commutative setting reduces to say that T t is superbounded with respect to m U if T t (L 2 (X, m)) ⊂ i 0 (L ∞ (X, m)) t > t 0 for some t 0 ≥ 0 and u L ∞ (X,m) ≤ v L 2 (X,m) whenever T t v = i 0 (u) for v ∈ L 2 (X, m), u ∈ L ∞ (X, m) and t > t 0 .In other words, T t is superbounded with respect to m U , if the induced Markovian semigroup satisfies In case (X, m) is an atomic measured space, the classical definition of super or ultracontractivity typically trivializes (see [15] Section 2.1): this happens, for example, if m is the counting measure because of the contractive embedding L 2 (X, m) ⊆ L ∞ (X, m).Superboundedness however may still be non trivial.
Let (X, m) be a countable, atomic measured space and let m = e −h m 0 for some function h and the counting measure m 0 .To simplify notations, we assume that e −U L 1 (X,m) = 1.For a fixed nonnegative measurable function V : X → [0, +∞) let us consider the semigroup T t : L 2 (X, m) → L 2 (X, m) T t v := e −tV v t ≥ 0 which is clearly Markovian with respect to the probability measure m U .
Lemma 7.2.The semigroup T t is supercontractive with respect to m U if and only if More precisely, T t extends to a contraction from L 2 (X, m U ) to L 2 (X, m) if and only if In case m is the counting measure we have t 0 = U/V ∞ /2.
is the quadratic form of an M -bimodule derivations (d λ Y , D(d λ Y )) on the standard bimodule L 2 (M ).In particular we show that in the Markovian case both (E λ Y , F λ Y ) and (d λ Y , D(d λ Y )) are represented by the symmetric embedding on L 2 (M ) of the unbounded, spatial derivations δ Y := i[Y, •] on M provided by the operator (Y, D(Y )) affiliated to M .In the subsequent Section 3, we prove natural lower bounds for the Dirichlet form (E λ Y , F λ Y ) in terms of the quadratic forms of the affiliated operators Y * Y , Y Y * , [Y, Y * ] and derive implications on the lower boundedness and discreteness of spectrum of (E λ

1 / 2 0
or, equivalently, of the Araki Hamiltonian ln ∆ 0 , we associate a completely Dirichlet form E λ Y choosing a densely defined, closed operator (Y, D(Y )) as in Lemma 2.1.For this choice there exists a canonical candidate, namely (Y 0 , D(Y 0 )).In general (E λ Y , F λ Y ) may depend upon the operator (Y, D(Y )) and not only upon the eigenvector ξ = Y ξ 0 it represents.The next result shows how this is connected to the GNS symmetry of the Markovian semigroup.Theorem 2.6.(GNS symmetry) Let (Y, D(Y )) be a densely defined, closed operator affiliated to

= 2 0
e −βN/2 ηe βN/2 η ∈ D(S 0 ).The modular group of ω β , satisfying the modular condition ω β (Aσω β −i (B)) = ω β (BA), for analytics elements A, B, is then given by σω β t = α −βt for t ∈ R.Regarding the Number Operator N as an operator affiliated to B(h) in its normal representation on L 2 (h) (i.e.acting, on a suitable domain of the Hilbert-Schmidt operators, by left composition), we have that the modular(Araki)  Hamiltonian is given by the strong sum of the densely defined, self-adjoint operators N and −JN J (belonging to commuting von Neumann algebras)− ln ∆ 0 = βN − JN J and its (discrete) spectrum is given by Sp(− ln ∆ 0 ) = βZ.Consequently Sp(∆ 1/) = e βZ/2 with uniform multiplicity one.Let us consider the annihilation and creation operators (A, D(A)), (A * , D(A * )) on h, defined on the domain D(A) := D( √ N ) =: D(A * ) asAe 0 := 0, Ae k := √ ke k−1 if k ≥ 1, A * e k := √ k + 1e k+1 k ∈ N.They satisfy the Canonical Commutation Relations AA * = A * A+I, as closed operators defined on D(N ), and allow to represent the Number Operator as N = A * A. All these operators and their functional calculi are understood as affiliated to B(h) acting by left composition on operators belonging to the Hilbert-Schmidt class L 2 (h).

7. 1 .
Generators of a class of positivity preserving semigroupsLet (A, D(A)) be a lower bounded, self-adjoint operator affiliated to a von Neumann algebra M and consider the C 0 -continuous, self-adjoint, positivity preserving semigroup on L 2 (M ), defined by T A t := e −tA j(e −tA ) = e −tA Je −tA J t ≥ 0.

7. 2 .
Superbounded semigroups on abelian atomic von Neumann algebras.The von Neumann algebra B(h) is atomic and this suggests to have a look at the superboundedness property in the abelian situation of atomic measured spaces.Let (X, m) be a locally compact, second countable, Hausdorff space, endowed with a fully supported Borel measure.Consider a real valued function U such that e −U ∈ L 1 (X, m) and define a probability measure by m U := e −U m X e −U dm.