abstract viewpoint on sch onberg correspondence: the...

Schonberg meets Haagerup

Abstract viewpoint on Schonberg correspondence:the Haagerup property via approximating

semigroups and their generatorsbased on joint work with M. Caspers, M. Daws, P. Fima and S. White

Adam Skalski

CNRS Besancon, IMPAN and University of Warsaw

Besancon, 18 December 2014

Schonberg and Haagerup for classical groups

Schonberg correspondence

G – discrete group.ϕ : G → C is positive definite if ϕ(e) = 1 and for all n ∈ N,g1, . . . , gn ∈ G , λ1, . . . , λn ∈ C

n∑i,j=1

ϕ(g−1i gj)λiλj ≥ 0.

ψ : G → C is conditionally positive definite if ψ(e) = 0 and for all n ∈ N,g1, . . . , gn ∈ G , λ1, . . . , λn ∈ C

n∑i=1

λi = 0 =⇒n∑

ψ(g−1i gj)λiλj ≥ 0.

Theorem (Schonberg correspondence)

A function ψ : G → C is conditionally positive definite if and only if foreach t ≥ 0 the function ϕt := etψ is positive definite.

Schonberg correspondence

G – discrete group.ϕ : G → C is positive definite if ϕ(e) = 1 and for all n ∈ N,g1, . . . , gn ∈ G , λ1, . . . , λn ∈ C

n∑i,j=1

ϕ(g−1i gj)λiλj ≥ 0.

ψ : G → C is conditionally positive definite if ψ(e) = 0 and for all n ∈ N,g1, . . . , gn ∈ G , λ1, . . . , λn ∈ C

n∑i=1

λi = 0 =⇒n∑

ψ(g−1i gj)λiλj ≥ 0.

Schonberg interpreted

Proof.

One direction is trivial, it is based by differentiating the family ϕt at 0.The other is a version of the GNS construction for conditionally positivedefinite functions.

This connects representation theory (positive definite functions) withgeometry (affine actions, cocycles, etc. – conditionally positive/negativedefinite functions)

Proof.

Equivalent definitions of the Haagerup property

A discrete group G has the Haagerup property (HAP) if the followingequivalent properties hold:

i G admits a mixing unitary representation which weakly contains thetrivial representation;

ii there exists a normalised sequence of positive definite functions ϕn

on G vanishing at infinity convergent to 1 pointwise;

iii there exists a real, proper, conditionally positive definite function ψon G ;

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

Equivalent definitions of the Haagerup property

A discrete group G has the Haagerup property (HAP) if the followingequivalent properties hold:

i G admits a mixing unitary representation which weakly contains thetrivial representation;

ii there exists a normalised sequence of positive definite functions ϕn

on G vanishing at infinity convergent to 1 pointwise;

iii there exists a real, proper, conditionally positive definite function ψon G ;

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

Equivalence (ii)⇐⇒(iii)

(iii)=⇒(ii) easy Schonberg correspondence(ii)=⇒(iii) defining

ψ(g) =∞∑n=1

αn(ϕn(g)− 1)

for some quickly convergent to 1 sequence of c0 functions ϕn andαn ∞.

Corollary

G has HAP iff it admits a continuous ‘semigroup’ ϕt = exp(tψ) ofpositive definite functions which are in c0.

ψ(g) =∞∑n=1

αn(ϕn(g)− 1)

Corollary

ψ(g) =∞∑n=1

αn(ϕn(g)− 1)

Corollary

Discrete quantum groups

Discrete quantum groups – general notations

G – a discrete quantum group, i.e.

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

i∈IMni ,equipped with the coproduct

∆ : `∞(G)→ `∞(G)⊗`∞(G)

carrying all the information about G

c0(G) =⊕

i∈IMni – the corresponding C∗-object

`2(G) – the GNS Hilbert space of the right invariant Haar weight on`∞(G)

G unimodular if the left and right weights coincide

∆ : `∞(G)→ `∞(G)⊗`∞(G)

c0(G) =⊕

∆ : `∞(G)→ `∞(G)⊗`∞(G)

c0(G) =⊕

Dual (compact) quantum groups

Each discrete quantum group G admits the dual compact quantum groupG.

L∞(G), C(G) – subalgebras of B(`2(G))L∞(G) has a canonical (Haar) normal state – tracial iff G is unimodular

Cu(G) – a ‘universal’ version of C(G)

Cu(G) contains a natural dense Hopf *-algebra, Pol(G); we have a

counit ε : Cu(G)→ C

states on Cu(G) ←→ states on Pol(G)

In particular for G – discrete group

L∞(G ) = VN(G )

C(G ) = C∗r (G ), Cu(G) = C∗(G )

L∞(G ) = VN(G )

C(G ) = C∗r (G ), Cu(G) = C∗(G )

L∞(G ) = VN(G )

C(G ) = C∗r (G ), Cu(G) = C∗(G )

Haagerup property for discrete quantum groups

Positive definite functions in the quantum world

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

via Bochner’s theorem, states on Cu(G) (i.e. states on ‘C∗(G)’);

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

L∞(G) (i.e. multipliers on ‘VN(G)’).

Actually they are equivalent.

Positive definite functions in the quantum world

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

via Bochner’s theorem, states on Cu(G) (i.e. states on ‘C∗(G)’);

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

L∞(G) (i.e. multipliers on ‘VN(G)’).

Actually they are equivalent.

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

Let G be a discrete quantum group. The following conditions areequivalent (and can be used as the definition of HAP):

G admits a mixing representation weakly containing the trivialrepresentation;

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

(equivalently, of states on Cu(G));

G admits a symmetric proper generating functional (i.e. a ‘realproper conditionally positive function’);

G admits a real proper cocycle.

Cpd functions, and convolution semigroups of states

Recall: states on Pol(G) ←→ states on Cu(G). As

Pol(G) = Linuαij : α ∈ Irr G, i , j = 1, . . . , nα,so functionals on Pol(G) ←→ families of matrices in Mnα :

µαi,j = µ(uαij )

Definition

A convolution semigroup of states on Pol(G) is a family (µt)t≥0 of states

on Pol(G) such that

i µt+s = µt ? µs := (µt ⊗ µs) ∆G, t, s ≥ 0;

ii µt(a)t→0+

−→ µ0(a) := ε(a), a ∈ Pol(G).

Convolution of functionals corresponds to multiplying matrices – for eachα ∈ Irr G there is a pointwise continuous semigroup of matrices (µαt )

Definition

on Pol(G) such that

ii µt(a)t→0+

−→ µ0(a) := ε(a), a ∈ Pol(G).

Definition

on Pol(G) such that

ii µt(a)t→0+

−→ µ0(a) := ε(a), a ∈ Pol(G).

Conditionally positive definite functions versusgenerating functionals

The following theorem is essentially due to M. Schurmann.

Theorem (Quantum Schonberg correspondence)

Each convolution semigroup of states on Pol(G) possesses a generating

functional L : Pol(G)→ C:

L(a) = limt→0+

µt(a)− ε(a)

t, a ∈ Pol(G).

The functional L is selfadjoint, vanishes at 1 and is positive on the kernelof the counit; in turn each functional enjoying these properties generatesa convolution semigroup of states.

Thus – conditionally positive definite functions on G correspond togenerating functionals on Pol(G).

Conditionally positive definite functions versusgenerating functionals

The following theorem is essentially due to M. Schurmann.

L(a) = limt→0+

µt(a)− ε(a)

t, a ∈ Pol(G).

The functional L is selfadjoint, vanishes at 1 and is positive on the kernelof the counit; in turn each functional enjoying these properties generatesa convolution semigroup of states.

Thus – conditionally positive definite functions on G correspond togenerating functionals on Pol(G).

Proof of the quantum Schonberg correspondence

L(a) = limt→0+

µt(a)− ε(a)

t, a ∈ Pol(G).

The functional L is selfadjoint, vanishes at 1 and is positive on thekernel of the counit; in turn each functional enjoying these propertiesgenerates a convolution semigroup of states.

Proof.

One direction is just differentiating µt at 0 (note that the derivativeexists!). The other is again a GNS construction + an application ofquantum stochastic processes of Hudson and Parthasarathy.

Proof of the quantum Schonberg correspondence

L(a) = limt→0+

µt(a)− ε(a)

t, a ∈ Pol(G).

The functional L is selfadjoint, vanishes at 1 and is positive on thekernel of the counit; in turn each functional enjoying these propertiesgenerates a convolution semigroup of states.

Proof.

One direction is just differentiating µt at 0 (note that the derivativeexists!). The other is again a GNS construction + an application ofquantum stochastic processes of Hudson and Parthasarathy.

HAP and convolution semigroups of states

Theorem (DFWS)

Let G be a discrete quantum group. The following are equivalent:

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

ii G admits a symmetric proper generating functional.

Here symmetric means that each matrix Lα ∈ Mnα , α ∈ IrrG, isself-adjoint, and proper that for α→∞ we have Lα −∞.

HAP and convolution semigroups of states

Theorem (DFWS)

Here symmetric means that each matrix Lα ∈ Mnα , α ∈ IrrG, isself-adjoint, and proper that for α→∞ we have Lα −∞.

Proof of the equivalence – symmetrizing ideas +Schonberg

Theorem (DFWS)

Proof.

(ii)=⇒(i) easy Schonberg correspondence(i)=⇒(ii) first show that the approximate unit built of ‘positive definite c0

functions’ on G (i.e. suitable states on Pol(G)) can be chosen invariantunder both the scaling automorphism group and the unitary antipode R(i.e. µαn are self-adjoint matrices). Then repeat the classical proof.

Theorem (DFWS)

Proof.

Theorem (DFWS)

Proof.

Semigroup corollary

Theorem (DFWS)

i G has HAP (i.e. c0(G) admits an approximate unit built of positivedefinite functions);

Corollary

G has HAP if and only if there exists a convolution semigroup of states(µt)t≥0 on Pol(G) built of ‘c0-functions’.

When G is unimodular, one can make in addition the matrices Lα scalar(a centrality property).

Semigroup corollary

Theorem (DFWS)

i G has HAP (i.e. c0(G) admits an approximate unit built of positivedefinite functions);

Corollary

G has HAP if and only if there exists a convolution semigroup of states(µt)t≥0 on Pol(G) built of ‘c0-functions’.

When G is unimodular, one can make in addition the matrices Lα scalar(a centrality property).

Aside on Property (T) for discrete quantum groups

Theorem (Kyed)

A unimodular G does not have Kazhdan property (T ) iff it admits a

generating functional L which is unbounded on Pol(G).

Proposition (DSW)

A unimodular G does not have Kazhdan property (T ) iff it admits agenerating functional L such that the matrices Lα are self-adjoint and thefamily of spectra of Lα is unbounded from below.

When can we add here the assumption that Lα is scalar?Important for understanding quantum T (1,1)!

Theorem (Kyed)

Proposition (DSW)

Theorem (Kyed)

Proposition (DSW)

HAP for tracial von Neumann algebras

Haagerup approximation property for finite vonNeumann algebras

A vNa M with a faithful normal tracial state τ has the von Neumannalgebraic Haagerup approximation property if there exists a net ofcompletely positive, τ -reducing, normal maps (Φi )i∈I on M such thatthe GNS-induced maps Ti on L2(M, τ) are compact and the net (Ti )i∈Iconverges 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– so the vNa HAP is a property of a (finite) von Neumann algebra. Themaps Φi in the definition of the vNa HAP can be chosen Markov – i.e.unital and trace preserving.

Haagerup approximation property for finite vonNeumann algebras

A vNa M with a faithful normal tracial state τ has the von Neumannalgebraic Haagerup approximation property if there exists a net ofcompletely positive, τ -reducing, normal maps (Φi )i∈I on M such thatthe GNS-induced maps Ti on L2(M, τ) are compact and the net (Ti )i∈Iconverges 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– so the vNa HAP is a property of a (finite) von Neumann algebra. Themaps Φi in the definition of the vNa HAP can be chosen Markov – i.e.unital and trace preserving.

Classical HAP via the approximation property for thevon Neumann algebra

Theorem ( Choda)

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

Proof.

If Γ has HAP, we have ‘good’ positive definite functions, so we can usethem to construct good ‘Herz-Schur’ multipliers on VN(Γ), which areL2-compact and converge to identity 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 ( Choda)

Proof.

ϕ(γ) = τ(Φ(λγ−1 )λγ), γ ∈ Γ,

Theorem ( Choda)

Proof.

ϕ(γ) = τ(Φ(λγ−1 )λγ), γ ∈ Γ,

Quantum group HAP via the approximation propertyfor the vNa

Recall that if G is unimodular, then the Haar state of G is a trace (in

particular, L∞(G) is a finite von Neumann algebra).

Theorem (DFSW, see also J.Kraus + Z.-J.Ruan)

Let G be a discrete unimodular quantum group. Then G has HAP if andonly if L∞(G) has the von Neumann algebraic Haagerup approximationproperty.

Proof.

Follows the classical idea of Choda: if G has HAP, we have good positivedefinite functions, so constructing multipliers out of them (see M.Junge+ M.Neufang + Z.J.Ruan, later also M.Daws) yields the approximationproperty for L∞(G) (this does not use the unimodularity).The other direction is based on ‘averaging’ approximating maps onL∞(G) into multipliers. Here unimodularity seems crucial.

Proof.

vNa HAP for general von Neumann algebras

Quantum group HAP via the approximation propertyfor the vNa revisited

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

Theorem

Let G be a discrete quantum group with the Haagerup property. LetM = L∞(G). There exists a net of Markov maps (Φi )i∈I on M such thateach of the respective GNS-induced maps Ti on `2(G) ≈ L2(M, h) iscompact and the net (Ti )i∈I converges to IL2(M,h) strongly. Moreoverone can choose Φi commuting with the action of the modular group.

von Neumann algebraic HAP for arbitrary vNa

Definition (M.Caspers + AS)

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

Independence of the choice of weight

Theorem (CS)

The Haagerup property does not depend on the choice of a faithfulnormal semifinite weight.

Idea of the proof:

show that in the state case can always make the approximationsuniformly bounded;

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 themodular action.

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 approachto the Haagerup property based on the standard form of the algebra M(the approximating maps in their approach act on the Hilbert space).

Theorem (COST)

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

Standard form approach

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

Theorem (COST)

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

KMS-induced maps

The 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 a normal completely positive, ϕ-reducing map. ItsKMS-implementation on L2(M, ϕ) is (informally!) given by the formula

TKMS(Ω12ϕxΩ

12ϕ) = Ω

12ϕΦ(x)Ω

The Haagerup property is equivalent to the KMS Haagerup property:

Definition

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

i on L2(M, ϕ) are compact and the net(TKMS

i )i∈I converges to IL2(M,ϕ) strongly.

KMS-induced maps

The 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 a normal completely positive, ϕ-reducing map. ItsKMS-implementation on L2(M, ϕ) is (informally!) given by the formula

TKMS(Ω12ϕxΩ

12ϕ) = Ω

12ϕΦ(x)Ω

The Haagerup property is equivalent to the KMS Haagerup property:

Definition

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

i on L2(M, ϕ) are compact and the net(TKMS

i )i∈I converges to IL2(M,ϕ) strongly.

Modularity

The KMS– and GNS–induced maps TKMS and T coincide if the map Φin question commutes with the modular group. If we can find suchapproximations, we say that M has a modular Haagerup property.

Markov property

The next result is surprisingly rather technical.

Theorem (CS)

Suppose that M has the Haagerup property and ϕ is a faithful normalstate on M. Then one can choose the approximating (in the KMS-sense)maps to be Markov and KMS-symmetric (i.e. theirKMS-implementations are selfadjoint operators on L2(M, ϕ)).

vNa HAP and Schonberg

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) = xσ-weakly. It is KMS-symmetric if each Φt is KMS symmetric, andimmediately L2-compact if each of the maps ΦKMS

t with t > 0 iscompact.

Theorem (CS, based on P.Jolissaint and F.Martin)

The following are equivalent:

i (M, ϕ) has the Haagerup property;

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

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) = xσ-weakly. It is KMS-symmetric if each Φt is KMS symmetric, andimmediately L2-compact if each of the maps ΦKMS

t with t > 0 iscompact.

Theorem (CS, based on P.Jolissaint and F.Martin)

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

vNa HAP via semigroups – the ideas of the proof

This does not go directly via Schonberg – we are not building thegenerator from the approximations!

start from a ‘good’ approximating sequence (Φn)n∈N and pass tothe Hilbert space picture, getting (Tn)n∈N

use the resolvent type tricks to show that one may assume that Tn

commute – write formulas of the type

θn :=1

n(T1 + · · ·+ Tn),∆n := n(I − θn),Rn,λ := λ(λI + ∆n)−1,

ρλ = limn→∞

Rn,λ−1 , Tn = ρ 1n

play a similar game once again to obtain a one-parameter semigroup

go back to the vNa, checking that all the properties we want arepreserved.

θn :=1

ρλ = limn→∞

Rn,λ−1 , Tn = ρ 1n

θn :=1

ρλ = limn→∞

Rn,λ−1 , Tn = ρ 1n

θn :=1

ρλ = limn→∞

Rn,λ−1 , Tn = ρ 1n

Modular HAP revisited

The following is now based on an observation of S. Neshveyev (and earlierremarks of Okayasu and Tomatsu).

Corollary

Let M be a von Neumann algebra with HAP and let ϕ be a faithfulnormal state on M. Then (M, ϕ) has the modular HAP if and only if ϕ isalmost periodic.

vNa HAP via Dirichlet forms

The generators of the approximating semigroup can be described byquantum Dirichlet forms – a very abstract form of conditionallypositive definite functions!

Theorem

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

Q(ξ) =∞∑n=1

λn|〈en, ξ〉|2, ξ ∈ DomQ,

where DomQ = ξ ∈ Hϕ :∑∞

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

Explicit examples – O+N

G = O+N

(based on the article of Fabio Cipriani, Uwe Franz and Anna Kula)

Proposition

There is an explicit decomposition `2(G) into finite-dimensional subspacesHd such that following formula defines a completely conservativeDirichlet form on `2(G), satisfying the conditions in the last theorem

Q(ξ) =∞∑s=1

Us′(N)

Us(N)‖PHd

ξ‖2, ξ ∈ DomQ.

Us(N) - Tchebyshev polynomials of the second kind

Explicit examples – SUq(2)

G = SUq(2)(based on the article of Kenny De Commer, Amaury Freslon and MakotoYamashita)

Proposition

There is an explicit decomposition `2(G) into finite-dimensional subspacesHd such that the following formula defines a completely conservativeDirichlet form on `2(G), satisfying the conditions in the last theorem

Q(ξ) =∞∑d=1

(q − q−1)ln q Aq,d‖PHd

ξ‖2, ξ ∈ DomQ,

Aq,d :=1

qd+1 − q−(d+1)

((d + 2)(qd − q−d) + d(q−(d+2) − qd+2)

Perspectives

The classical Schonberg correspondence connects representation theorywith geometry.The same is true for

discrete quantum groups (generating functionals −→ cocycles!)

even general von Neumann algebras (Dirichlet forms −→ Diracoperators!)

This opens many new questions, for example related to property (T).

Perspectives

References

HAP for locally compact quantum groups:

M.Daws, P.Fima, A.S. and S.White, Haagerup property for locally compactquantum groups, Crelle, 2012.

This talk:

M.Caspers and A.S., The Haagerup property for arbitrary von Neumannalgebras, IMRN, 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, CMP 2014.

abstract viewpoint on sch onberg correspondence: the...

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