The(∗∗)-Haagerup Property for C∗-Algebras∗

Changjing LI Xiaochun FANG

1 Introduction

Recall that a discrete group Γ is said to have the Haagerup property if there is a net{Φn}of positive definite functions,each of which vanishes at infinity and the net converges to 1 point wise.This definition is motivated by the work of Haagerup[8]where he proved that free groups have such a property.It is well known that if we replace the condition“vanishes at infinity” in the above definition by “has a finite support”,then this is equivalent to amenability.Hence,the Haagerup property is considered as a weak version of amenability.The Haagerup property has been intensively studied in the literature.On the one hand,the Haagerup property is satisfied by many important non-amenable groups,including free groups,the Coxeter groups,and so on.It is also known that the Haagerup property is equivalent to several other important properties,such as the a-T-menabilty introduced by Gromov[7].On the other hand,the groups with the Haagerup property do not have the(relative)property(T),which is a rigidity property of discrete groups.In many situations,a group with the(relative)property(T)is essentially hard to study,so at least for the group with the Haagerup property,essential difficulties would not arise.We refer to the book[3]for a comprehensive account of this subject.

A similar Haagerup property has been considered for finite von Neumann algebras in[4–5,9,11].Suppose that M is a finite von Neumann algebra with a normal faithful tracial state τ.We say that M has the Haagerup property with respect to τ(shortly,M has the Haagerup property for von Neumann algebras)if there is a net{Φi}i∈Iof unital completely positive normal maps from M to itself such that(1)τ◦Φi≤ τ,(2)each Φidefines a compact operator?Φion L2(M,τ),and(3)?Φi(A)−A?τ=τ((Φi(A)−A)∗(Φi(A)−A)for all A∈M.We note that condition(1)in the definition guarantees that each Φidefines a bound linear mapon L2(M,τ).Jolissaint[9]proved that the definition does not depend on the choice of faithful normal tracial states.It was shown by Choda[4]that a discrete group Γ has the Haagerup property if and only if its group von Neumann algebra L(Γ)has the Haagerup property for von Neumann algebras.

Recently,Dong[6]introduced a notion of the Haagerup property for unital C∗-algebras admitting a faithful tracial state,by imitating the case of von Neumann algebras.The definition of the Haagerup property for unital C∗-algebras is the same as above except for the assumption that τ is only a faithful tracial state and{Φi}i∈Iare only unital completely positive maps.Dong[6]showed that a discrete group Γ has the Haagerup property if and only if its reduced group C∗-algebra C∗r(Γ)has the Haagerup property with respect to the canonical tracial state.Suzukai[13]provided an example,which showed that the Haagerup property for C∗-algebras strictly depends on the choice of faithful tracial states.He also proved that a nuclear C∗-algebra with a faithful tracial state always has the Haagerup property for C∗-algebras.

Because the Haagerup property for C∗-algebras strictly depends on the choice of faithful tracial states,this paper is an attempt to find a hopefully better definition of the Haagerup property for C∗-algebras by looking at the Haagerup property for the biduals of C∗-algebras(such property is thus called the(∗∗)-Haagerup property)such that such a definition does not depend on the choice of faithful tracial states.In this paper,we will show that a nuclear unital C∗-algebra with a faithful tracial state always has the(∗∗)-Haagerup property.We also show,if Γ is an amenable countable discrete group and unital separable C∗-algebra A has the(∗∗)-Haagerup property,then the reduced crossed product A?α,rΓ has the(∗∗)-Haagerup property.Moreover,we will give an answer to the following open question(see[13,Question 6.2]):

Does the Haagerup property for C∗-algebras pass to a C∗-subalgebra?That is,let C∗-algebra A have the Haagerup property with respect to a faithful tracial state τ,and B be a C∗-subalgebra of A.Then does B have the Haagerup property with respect to τ|B?

2 The(∗∗)-Haagerup Property for C∗-Algebras

definition 2.1 Let A be a unital C∗-algebra and τ be a faithful tracial state on A.We say that A has the(∗∗)-Haagerup property with respect to τ(shortly,A has the(∗∗)-Haagerup property)if A∗∗has the Haagerup property for von Neumann algebras.

Remark 2.1(1)A∗∗is the bidual(the universal enveloping von Neumann algebra)of A,and we refer the readers to the books[10]and[14].

(2)If τ is a faithful tracial state on A,then the biduals map τ∗∗:A∗∗→ C∗∗=C is a normal faithful tracial state on A∗∗.

(3)Since the Haagerup property for von Neumann algebras does not depend on the choice of faithful normal tracial states,our definition of the(∗∗)-Haagerup property for C∗-algebras does not depend on the choice of faithful tracial states.

Suzuki[13]gave an in finite-dimensional example of C∗-algebra which has both the Haagerup property with respect to some faithful tracial state and the property(T).

Example(see[13,Example 4.15])Let n≥3,and on the group algebra C[SLn(Z)]of SLn(Z),define the C∗-seminorm?x?finas follows:

Then define the C∗-algebraas the completion of C[SLn(Z)]with respect to the norm? ·?fin.Suzuki[13,Theorem 4.18]proved that there exist two faithful tracial states τ1and τ2such thathas the Haagerup property with respect to τ1anddoes not have the Haagerup property with respect to τ2.Hence the Haagerup property for C∗-algebras does depend on the choice of faithful tracial states.

Remark 2.2(1)If a unital C∗-algebra A has the Haagerup property with respect to faithful tracial state τ,then A has the(∗∗)-Haagerup property.Indeed,if there exists a net of unital completely positive maps Φi:A → A satisfying three conditions in the definition of the Haagerup property for C∗-algebras,then the biduals mapsare unital completely positive normal maps,and it is easy to prove thatsatisfy three conditions in the definition of the Haagerup property for von Neumann algebras.Hence A∗∗has the Haagerup property with respect to normal faithful tracial state τ∗∗,that is,A has the(∗∗)-Haagerup property.But if A has the(∗∗)-Haagerup property,then A does not have the Haagerup property with some faithful tracial state in general.Indeed,by[13,Theorem 4.18]there exist two faithful tracial states τ1and τ2such thathas the Haagerup property with respect to τ1(hencehas the(∗∗)-Haagerup property)anddoes not have the Haagerup property with respect to τ2.

(2)It is well known(see[9,Theorem 2.3])that the Haagerup property for von Neumann algebras can pass to every von Neumann subalgebra.Hence,for von Neumann algebra M,if M has the(∗∗)-Haagerup property,then M also has Haagerup property for von Neumann algebras.By Remark 2.2(1),M has the(∗∗)-Haagerup property if and only if M has the Haagerup property for von Neumann algebras.

(3)Let A be a unital C∗-algebra with a faithful tracial state τ.If A has the(∗∗)-Haagerup property and there exists a τ-preserving conditional expectation E from A∗∗to A,then A has the Haagerup property with respect to τ.Indeed,since A has the(∗∗)-Haagerup property,there exists a net{Φi}i∈Iof unital completely positive normal maps on A∗∗satisfying three conditions in the definition of the Haagerup property for von Neumann algebras.Let Ψi=E◦Φi|A.Then it is easy to prove that{Ψi}i∈Iis a net of unital completely positive maps on A satisfying three conditions in the definition of the Haagerup property for C∗-algebras.Hence A has the Haagerup property with respect to τ.

In 2013,Suzuki gave the following open question.

Question 2.1(see[13,Question 6.2])Does the Haagerup property for C∗-algebras pass to a C∗-subalgebra?That is,let C∗-algebra A have the Haagerup property with respect to a faithful tracial state τ,and B be a C∗-subalgebra of A.Then does B have the Haagerup property with respect to τ|B?

Note first that this is true if A is nuclear(see[13,Corollary 3.7]).Note also that Question 2.1 has a positive answer in the context of the von Neumann algebras.The reason why we can prove this for von Neumann algebras is that we can always construct a trace-preserving condition expectation(see[2,Lemma 1.5.11]).But in the context of the C∗-algebras and,we can not construct a condition expectation in general,even if we do not consider the condition about the trace.For example,let A be a nuclear C∗-algebra,and B be a C∗-subalgebra of A which is not nuclear.Then there is no condition expectation from A to B,because any range of a conditional expectation on a nuclear C∗-algebra is nuclear.

Now we give an example,which shows that Question 2.1 does not have a postive answer in general.By[13,Theorem 4.18]there exist two faithful tracial states τ1and τ2such thathas the Haagerup property with respect to τ1anddoes not have the Haagerup property with respect to τ2.By Remark 2.2(1),we have thathas the Haagerup property with respect to.Since the Haagerup property for von Neumann algebras does not depend on the choice of faithful normal tracial states,sohas the Haagerup property with respect tois a C∗-subalgebra ofdoes not have the Haagerup property with respect to

3 Some Main Results

Theorem 3.1 Let A be a unital nuclear C∗-algebra and τ be a faithful tracial state on A.Then A has the(∗∗)-Haagerup property.

Proof Since the(∗∗)-Haagerup property for C∗-algebras is weaker than the Haagerup property for C∗-algebras(see Remark 2.2(1)),by[13,Theorem 3.6]it is easy to get the theorem.Now we give a simple proof.It is mentioned in[9]that the injective finite von Neumann algebras have the Haagerup property for von Neumann algebras.Indeed,it is a deep and by now classical result that injective von Neumann algebras are semidiscrete.It then follows from[12,Proposition 4.6]that injective von Neumann algebras which admit a faithful normal trace state have the Haagerup property for von Neumann algebras.It is well known that A is nuclear if and only if A∗∗is injective(see[1,IV.3.1.5]).The proof of the theorem is finished.

In a way similar to[13,Corollary 3.7],we also have the following corollary.

Corollary 3.1(1)Let A be a unital exact C∗-algebra with a faithful amenable tracial state.Then A has the(∗∗)-Haagerup property.

(2)Let A be a unital residually finite-dimensional C∗-algebra with a faithful tracial state.Then A has the(∗∗)-Haagerup property.

Now we gather some heredity results concerning the(∗∗)-Haagerup property.

Theorem 3.2 Let A be a unital C∗-algebra and τ be a faithful tracial state on A.Let B and C be unital C∗-algebras.Assume that B and C have the(∗∗)-Haagerup property.

(1)If 1A∈ B is a C∗-subalgebra of A,and A has the(∗∗)-Haagerup property,then B has the(∗∗)-Haagerup property.

(2)Assume that there exists a sequence of unital C∗-algebras An,each of which has the(∗∗)-Haagerup property with respect to faithful tracial states τn,and assume that for every n there exist unital completely positive maps Sn:A→Anand Tn:An→A such that τn◦Sn≤ τ,τ◦Tn≤ τn,and such that?Tn◦Sn(A)−A?τ→ 0 as n→ ∞.Then A has the(∗∗)-Haagerup property.

(3)If A has the(∗∗)-Haagerup property,then for each n∈N,Mn(A)has the(∗∗)-Haagerup property.

(4)The spatial tensor product B⊗C has the(∗∗)-Haagerup property.

(5)The direct sum B⊕C has the(∗∗)-Haagerup property.

Proof(1)and(2)follow from[9,Theorem 2.3(i)and(ii)].By[6,Lemma 2.4]and the fact(Mn(A))∗∗=Mn(A∗∗),we get(3).By[9,Theorem 2.3(iii)]and the fact(B⊗C)∗∗=B∗∗⊗C∗∗,where⊗is the von Neumann algebra tensor product,we can prove(4).By[13,Theorem 3.12(1)]and the fact(B⊕C)∗∗=B∗∗ ⊕C∗∗,we can prove(5).

Theorem 3.3LetΓbe an amenable countable discrete group and A be a unital separable C∗-algebra with a faithful tracial state τ.If A has the(∗∗)-Haagerup property,then the reduced crossed product A?α,rΓhas the(∗∗)-Haagerup property,where αis a τ-preserving action ofΓ.

Proof Since Γ is amenable,there exists a sequence of finite setsFn⊆Γ such that

for all finite setsE⊆Γ.We defineϕn:A?α,rΓ→A⊗MFn(C)such that

andψn:A⊗MFn(C)→A?α,rΓ such that

Dong[6,Theorem 2.5]showed thatϕn,ψnare unital completely positive maps such thatτn◦ϕn≤τ?,τ?◦ψn≤τn,whereτnandτ?are the induced traces ofτonA⊗MFn(C)andA?α,rΓ,respectively,and such that?ψn◦ϕn(x)−x?τ?→0 for allx∈A?α,rΓ.SinceAhas the(∗∗)-Haagerup property,it follows from Theorem 3.2(3)thatA⊗MFn(C)has the(∗∗)-Haagerup property.By Theorem 3.2(2),we get thatA?α,rΓ has the(∗∗)-Haagerup property.

Acknowledgements The authors would like to thank the referee for his valuable comments and suggestions.They would also like to thank Professor Genkai Zhang for the numerous discussions on the subject.

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