haagerup property for arbitrary von neumann algebras or on...

Haagerup property for arbitrary von Neumann algebrasor on inspirations coming from quantum groups

based on joint work with M. Caspers,building on earlier work with M. Daws, P. Fima and S. White

related to the work of R. Okayasu and R. Tomatsu

Adam Skalski

IMPAN and University of Warsaw

Cheongpung, 11th August 2014

Adam Skalski (IMPAN & UW) Haagerup property for vNas Cheongpung, 11th August 2014 1 / 26

Equivalent definitions of the Haagerup property

A discrete group G has the Haagerup property (HAP) if the following equivalentproperties hold:

there exists a normalised sequence of positive definite functions on Gvanishing at infinity convergent to 1 pointwise;

G admits a mixing unitary representation which weakly contains the trivialrepresentation;

there exists a real, proper, conditionally negative definite function on G ;

G admits a proper affine action on a real Hilbert space.

Basic properties and examples

amenable groups have HAP;

G has both HAP and property (T ) if and only if G is compact;

free groups, finitely generated Coxeter groups have HAP;

SL(2,Z) has HAP.

Haagerup approximation property for finite von Neumannalgebras

A vNa M with a faithful normal tracial state τ has the von Neumann algebraicHaagerup approximation property if there exists a net of completely positive,τ -reducing, normal maps (Φi )i∈I on M such that the GNS-induced maps Ti onL2(M, τ) are compact and the net (Ti )i∈I converges to IL2(M,τ) strongly.

L2(M, τ) – the GNS Hilbert space of the pair (M, τ)

Ti (xΩτ ) = Φi (x)Ωτ , x ∈ M.

P.Jolissaint showed that this property does not depend on the choice of τ – sothe vNa HAP is a property of a (finite) von Neumann algebra.

Haagerup approximation property for finite von Neumannalgebras

A vNa M with a faithful normal tracial state τ has the von Neumann algebraicHaagerup approximation property if there exists a net of completely positive,τ -reducing, normal maps (Φi )i∈I on M such that the GNS-induced maps Ti onL2(M, τ) are compact and the net (Ti )i∈I converges to IL2(M,τ) strongly.

L2(M, τ) – the GNS Hilbert space of the pair (M, τ)

Ti (xΩτ ) = Φi (x)Ωτ , x ∈ M.

P.Jolissaint showed that this property does not depend on the choice of τ – sothe vNa HAP is a property of a (finite) von Neumann algebra.

HAP for finite von Neumann algebras – continued

The maps Φi in the definition of the vNa HAP can be chosen Markov – i.e. unitaland trace preserving. For Markov maps strong convergence of the GNSimplementations = pointwise σ-weak convergence of the original maps.

Theorem (F.Boca)

The free product of finite von Neumann algebras with vNa HAP (with respect tocorresponding tracial states), possibly with amalgamation over afinite-dimensional subalgebra, has HAP.

HAP for finite von Neumann algebras – continued

The maps Φi in the definition of the vNa HAP can be chosen Markov – i.e. unitaland trace preserving. For Markov maps strong convergence of the GNSimplementations = pointwise σ-weak convergence of the original maps.

Theorem (F.Boca)

The free product of finite von Neumann algebras with vNa HAP (with respect tocorresponding tracial states), possibly with amalgamation over afinite-dimensional subalgebra, has HAP.

Classical HAP via the approximation property for the vonNeumann algebra

Theorem (M. Choda)

A discrete group Γ has HAP if and only if VN(Γ) has the von Neumann algebraicHaagerup approximation property.

Proof.If Γ has HAP, we have ‘good’ positive definite functions, so we can use them toconstruct ucp Schur multipliers on VN(Γ), which are L2-compact and converge toidentity pointwise σ-weakly.The other direction is based on ‘averaging’ approximating maps on intomultipliers VN(Γ): more specifically, defining

ϕ(γ) = τ(Φ(λγ−1)λγ), γ ∈ Γ,

yields ‘good’ positive definite functions.

Theorem (M. Choda)

ϕ(γ) = τ(Φ(λγ−1)λγ), γ ∈ Γ,

Theorem (M. Choda)

ϕ(γ) = τ(Φ(λγ−1)λγ), γ ∈ Γ,

Example of an application

Corollary

If Γ1, Γ2 are discrete groups with HAP, then Γ1 ? Γ2 has HAP; similarly Γ1 ?H Γ2 ifH is a finite subgroup.

Discrete quantum groups – general notations

G – a discrete quantum group a la Woronowicz, i.e.

`∞(G) – a von Neumann algebra, which is of the form∏

i∈IMni , equipped withthe coproduct

∆ : `∞(G)→ `∞(G)⊗`∞(G)

carrying all the information about G

c0(G) =⊕

i∈IMni – the corresponding C∗-object

∆ : `∞(G)→ `∞(G)⊗`∞(G)

c0(G) =⊕

∆ : `∞(G)→ `∞(G)⊗`∞(G)

c0(G) =⊕

Positive definite functions in the quantum world

What should positive definite functions on G be? There are at least two possiblepoints of view:

via Bochner’s theorem, states on states on C∗(G);

elements in `∞(G) yielding ‘completely positive multipliers’ on VN(G).

See a recent JFA paper by M.Daws and P.Salmi.

Positive definite functions in the quantum world

What should positive definite functions on G be? There are at least two possiblepoints of view:

via Bochner’s theorem, states on states on C∗(G);

elements in `∞(G) yielding ‘completely positive multipliers’ on VN(G).

See a recent JFA paper by M.Daws and P.Salmi.

Haagerup property for discrete quantum groups

Theorem (M.Daws, P.Fima, S.White, AS)

Let G be a discrete quantum group. The following conditions are equivalent (andcan be used as the definition of HAP):

c0(G) admits an approximate unit built of ‘positive definite functions’;

G admits a mixing representation weakly containing the trivialrepresentation;

the dual quantum group G admits a symmetric proper generating functional(a ‘conditionally negative definite function on G’);

G admits a real proper cocycle (‘a part of an affine action of G on a realHilbert space’).

Adam Skalski (IMPAN & UW) Haagerup property for vNasCheongpung, 11th August 2014 10 /

Quantum group HAP via the approximation property forthe vNa

If G is unimodular, then the Haar state of G is a trace (in particular, VN(G) is afinite von Neumann algebra).

TheoremLet G be a discrete unimodular quantum group. Then G has HAP if and only ifVN(G) has the von Neumann algebraic Haagerup approximation property.

Proof.Follows the classical idea of Choda: if G has HAP, we have good positive definitefunctions, so constructing multipliers out of them (see M.Junge + M.Neufang +Z.J.Ruan, later also M.Daws) yields the approximation property for VN(G) (thisdoes not use the unimodularity).The other direction is based on ‘averaging’ approximating maps on VN(G) intomultipliers. Here unimodularity seems crucial.

Quantum group HAP via the approximation property forthe vNa revisited

What if a discrete quantum group G is not unimodular?The Haar state h on VN(G) is no longer tracial. But...

Theorem

Let G be a discrete quantum group with the Haagerup property. Let M = VN(G).There exists a net of normal completely positive, unital, h-preserving maps(Φi )i∈I on M such that each of the respective GNS-induced maps Ti on`2(G) ≈ L2(M, h) is compact and the net (Ti )i∈I converges to IL2(M,h) strongly.Moreover one can choose Φi commuting with the action of the modular group.

von Neumann algebraic HAP for arbitrary vNa

Definition (DFWS)

Let (M, ϕ) be a von Neumann algebra with a faithful normal state. We say that(M, ϕ) has the Haagerup property if there exists a net of normal completelypositive, unital, ϕ-preserving maps (Φi )i∈I on M such that the GNS-inducedmaps Ti on L2(M, ϕ) are compact and the net (Ti )i∈I converges to IL2(M,ϕ)

strongly.

von Neumann algebraic HAP for arbitrary vNa – take II

Definition (M.Caspers + AS)

Let (M, ϕ) be a von Neumann algebra with a faithful normal state. We say that(M, ϕ) has the Haagerup property if there exists a net of normal completelypositive, ϕ-reducing maps (Φi )i∈I on M such that the GNS-induced maps Ti onL2(M, ϕ) are compact and the net (Ti )i∈I converges to IL2(M,ϕ) strongly.

von Neumann algebraic HAP for arbitrary vNa – take III

Definition (M.Caspers + AS)

Let (M, ϕ) be a von Neumann algebra with a faithful normal semifinite weight.We say that (M, ϕ) has the Haagerup property if there exists a net of normalcompletely positive, ϕ-reducing maps (Φi )i∈I on M such that the GNS-inducedmaps Ti on L2(M, ϕ) are compact and the net (Ti )i∈I converges to IL2(M,ϕ)

strongly.

Immediate questions

does the property depend on the choice of ϕ?

can one always get the approximating maps unital and ϕ-preserving?

are there any other possible choices?

Immediate questions

Answers, i.e. the magic of crossed product duality

Theorem (CS)

The Haagerup property does not depend on the choice of a faithful normalsemifinite weight.

Idea of the proof:

show that (M, ϕ) has HAP iff all its ‘nice’ corners have HAP;

prove that one can change weights if the algebra is semifinite;

prove that HAP is stable under passing to crossed products by(ϕ-preserving) actions of amenable groups;

use the Takesaki-Takai duality for the crossed products by the modularaction.

Theorem (CS)

Idea of the proof:

Theorem (CS)

Idea of the proof:

Theorem (CS)

Idea of the proof:

Theorem (CS)

Idea of the proof:

Standard form approach

At the same time R.Okayasu and R.Tomatsu developed another approach to theHaagerup property based on the standard form of the algebra M (theapproximating maps in their approach act on the Hilbert space).

Theorem (OT, COST)

A vNa M has the Haagerup property in the sense of CS if and only if it has theHaagerup property in the sense of OT.

Open questions

Standard form approach

At the same time R.Okayasu and R.Tomatsu developed another approach to theHaagerup property based on the standard form of the algebra M (theapproximating maps in their approach act on the Hilbert space).

Theorem (OT, COST)

A vNa M has the Haagerup property in the sense of CS if and only if it has theHaagerup property in the sense of OT.

Open questions

ModularityThe approach of OT is related to considering the KMS–induced maps.

Definition

Let (M, ϕ) be a vNa with a faithful normal semifinite weight, Φ : M → M anormal completely positive, ϕ-reducing map. Its KMS-implementation onL2(M, ϕ) is (informally!) given by the formula

TKMS(Ω12ϕxΩ

12ϕ) = Ω

12ϕΦ(x)Ω

Once again the crossed product technique yields the fact that the Haagerupproperty is equivalent to the KMS Haagerup property:

Definition

(M, ϕ) has the KMS Haagerup property if there exists a net of normal completelypositive, ϕ-reducing maps (Φi )i∈I on M such that the KMS-induced maps TKMS

on L2(M, ϕ) are compact and the net (TKMSi )i∈I converges to IL2(M,ϕ) strongly.

The KMS– and GNS–induced maps coincide if the map in question commuteswith the modular group. We do not know if one can always achieve it!

Definition

TKMS(Ω12ϕxΩ

12ϕ) = Ω

12ϕΦ(x)Ω

Definition

TKMS(Ω12ϕxΩ

12ϕ) = Ω

12ϕΦ(x)Ω

Definition

Markov property

The next result is surprisingly rather technical.

Theorem (CS, see also OT)

Suppose that M has the Haagerup property and ϕ is a faithful normal state onM. Then one can choose the approximating (in the KMS-sense) maps to beMarkov and KMS-symmetric (i.e. their KMS-implementations are selfadjointoperators on L2(M, ϕ)).

Corollary

Free product of vNas with faithful normal states which have the Haagerupproperty has the Haagerup property.

Markov property

The next result is surprisingly rather technical.

Theorem (CS, see also OT)

Suppose that M has the Haagerup property and ϕ is a faithful normal state onM. Then one can choose the approximating (in the KMS-sense) maps to beMarkov and KMS-symmetric (i.e. their KMS-implementations are selfadjointoperators on L2(M, ϕ)).

Corollary

Free product of vNas with faithful normal states which have the Haagerupproperty has the Haagerup property.

vNa HAP via semigroups(M, ϕ) – vNa with a faithful normal state

Definition

A Markov semigroup Φt : t ≥ 0 on (M, ϕ) is a semigroup of Markov maps on

M such that for all x ∈ M we have Φt(x)t→0+−→ Φ0(x) = x σ-weakly. It is

KMS-symmetric if each Φt is KMS symmetric, and immediately L2-compact ifeach of the maps ΦKMS

t with t > 0 is compact.

The next result was inspired by the theorem for finite von Neumann algebras dueto P.Jolissaint and F.Martin.

Theorem (CS)

The following are equivalent:

i (M, ϕ) has the Haagerup property;

ii there exists an immediately L2-compact KMS-symmetric Markov semigroupΦt : t ≥ 0 on M.

Open questions

vNa HAP via semigroups(M, ϕ) – vNa with a faithful normal state

Definition

A Markov semigroup Φt : t ≥ 0 on (M, ϕ) is a semigroup of Markov maps on

M such that for all x ∈ M we have Φt(x)t→0+−→ Φ0(x) = x σ-weakly. It is

KMS-symmetric if each Φt is KMS symmetric, and immediately L2-compact ifeach of the maps ΦKMS

t with t > 0 is compact.

The next result was inspired by the theorem for finite von Neumann algebras dueto P.Jolissaint and F.Martin.

Theorem (CS)

The following are equivalent:

ii there exists an immediately L2-compact KMS-symmetric Markov semigroupΦt : t ≥ 0 on M.

Open questions

vNa HAP via Dirichlet formsThe next result characterises the Haagerup property for a von Neumann algebrain terms of objects playing the role of the generators of the approximatingsemigroup – quantum Dirichlet forms. In the classical world these are related toconditionally negative definite functions.

TheoremThe following are equivalent:

ii L2(M, ϕ) admits an orthonormal basis (en)n∈N and a non-decreasingsequence of non-negative numbers (λn)n∈N such that limn→∞ λn = +∞ andthe prescription

Q(ξ) =∞∑n=1

λn|〈en, ξ〉|2, ξ ∈ DomQ,

where DomQ = ξ ∈ Hϕ :∑∞

n=1 λn|〈en, ξ〉|2 <∞, defines a conservativecompletely Dirichlet form.

Open questions

how to characterise the HAP for discrete non-unimodular G via the vonNeumann algebra L∞(G)?

is the modular vNa HAP equivalent to the usual HAP?

can the approximating maps in HAP be chosen so that they are Markoveven in the weight case?

how to phrase the vNa HAP in the language of correspondences (in thefinite case see Bannon+Fang and Peterson)?

Open questions

References

HAP for locally compact quantum groups:

M.Daws, P.Fima, A.S. and S.White, Haagerup property for locally compactquantum groups, Crelle, 2014.

This talk:

M.Caspers and A.S., The Haagerup property for arbitrary von Neumann algebras,preprint 2013.

M.Caspers, R.Okayasu, R.Tomatsu and A.S., Generalisations of the Haagerupapproximation property to arbitrary von Neumann algebras,C. R.Math. Acad. Sci. Paris, 2014.

M.Caspers and A.S., The Haagerup approximation property for von Neumannalgebras via quantum Markov semigroups and Dirichlet forms, preprint 2014.

haagerup property for arbitrary von neumann algebras or on...

Documents