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Comptes Rendus Mathématique
Volume 354, n° 1
pages 87-90 (janvier 2016)
Doi : 10.1016/j.crma.2015.10.013
Received : 10 July 2015 ;  accepted : 21 October 2015
KK-theory of A-valued semi-circular systems
KK-théorie des systèmes semi-circulaires A-valués

Emmanuel Germain a , Pierre Umber b
a LMNO UMR 6139, Université de Caen et CNRS, France
b ENS Lyon, France

 Abstract

We compute in this article the KK-theory of A-valued semi-circular systems thanks to tools developed by Pimsner (see [[1]]) to study generalized Toeplitz algebras.

 Résumé

On calcule dans cet article la KK-théorie de systèmes semi-circulaires A-valués à l'aide d'outils développés par Pimsner (voir [[1]]) pour étudier les algèbres de Toeplitz généralisées.

 Outline Masquer le plan

To begin with, we will need a result in Hilbert module theory.

 Proposition 0.1

Let A, C be  -algebras, B a sub-  -algebra of C, E a Hilbert module over A, and   a *-morphism. Let   be the inclusion,   and  . Then   is naturally endowed with a structure of Hilbert module over B and  .

Indeed, let  ,  . We have:
<∑xi⊗bi,∑xk′⊗bk′>=∑i,kbi⁎ϕ(<xi,xk′>)bk′∈B. As B is closed in C , we have:  . As a result,   is naturally endowed with a structure of pre-Hilbert module over B , which is complete because   is a closed subspace of the Hilbert module  .

For the second part of the proposition, let  . If  , then  . Then π induces  . We clearly have:
<π˜(∑xi⊗bi),π˜(∑xi⊗bi)>=<∑xi⊗bi,∑xi⊗bi>, so   is an isometry. As   is complete,   extends to an isometry   on  . As   is an isometry,   is closed in  , but   contains a dense subspace of  , so   is an isomorphism and  .

Let's turn now to our main result. Let A be a  -algebra with unit, and E be a Hilbert module over A with an isometric *-morphism   which endowed E with a left action. The algebra A is supposed to be separable and E countably generated. We will denote by   the Fock space associated to E , which is   (where  ). Each   is a left A -module, thus   is endowed with a diagonal left action over A .

Let   and   be the left creation operator  . Then   and the annihilation operator is given by  .

We denote by   the associated Toeplitz algebra, which is the  -algebra generated by A and the operators  .

If E is also endowed with an anti-linear involution   then there is a natural *-subalgebra of  , that we denote by  , and is generated by A and elements  . This algebra is mainly studied in a Von Neumann algebra context (see for example [[2]] and [[3]]). We will here compute its KK-theory as a particular case of the following theorem.

 Theorem 0.2

Let S be any sub-  -algebra of   which contains A and is generated by linear combinations of creation and annihilation operators. Then S is KK-equivalent to A

According to Pimsner (see Proposition 3.3 in [[1]]), Toeplitz algebras satisfy the following universal property:

 Proposition 0.3

Let B be a  -algebra and   a *-morphism. We suppose that there is a family   in B such that:

 1) is  -linear 2) and 3)
Then σ extends to a unique morphism on   such that  .

We denote by   the inclusion of A in S ,   the inclusion of S in   and  . Let P be the projection in   onto   and  . Let   given by the diagonal left action of  , and  . We also define  . Then   satisfies conditions in Proposition 0.3, so   extends to a representation   of  .

Let   where   is defined by  . Then β is an element of   (see Lemma 4.2 and Definition 4.3 in [[1]]).

We have the relations   and  , where  , for every  -algebra C , is the multiplicative unit in the ring   (see Theorem 4.4 in [[1]]). We consider  .

 Proposition 0.4

We have the relations   and  .

Indeed, for the first one we have  .

For the second one, we first recall all the tools which are introduced in Pimsner's article in the proof of Theorem 4.4. Let   be the operator such that, for  ,   acts on   by   and is equal to zero on   (note that   is a *-morphism). Let   be the operator such that, for  ,   acts on   by   and is equal to zero on  . Note that   on   and is equal to zero on   and  .

 Lemma 0.5

Consider   and  . We define
T˜ξ,t=defcos⁡(π2t)τ0(Tξ)+sin⁡(π2t)τ1(Tξ)+π1(Tξ)⊗1TE. The couple   satisfies the conditions in Proposition 0.3, and thus   extends to a representation  .

Conditions 1) and 2) are easy to check.

As regards condition 3), we have:   where
I=cos2⁡(π2t)(τ0(Tξ))⁎τ0(Tζ)+sin⁡(π2t)cos⁡(π2t)(τ0(Tξ))⁎τ1(Tζ)+cos⁡(π2t)(τ0(Tξ))⁎π1(Tζ)⊗1TE,J=cos⁡(π2t)sin⁡(π2t)(τ1(Tξ))⁎τ0(Tζ)+sin2⁡(π2t)(τ1(Tξ))⁎τ1(Tζ)+sin⁡(π2t)(τ1(Tξ))⁎π1(Tζ)⊗1TE,K=cos⁡(π2t)((π1(Tξ))⁎⊗1TE)τ0(Tζ)+sin⁡(π2t)((π1(Tξ))⁎⊗1TE)τ1(Tζ)+((π1(Tξ))⁎π1(Tζ))⊗1TE). Then we compute each term on the subspace where it doesn't vanish. Remark that the subspaces   of   are stable for  . Let  ,  . We have:

 • ; • ; • ; • ; • ; • ; • ; • .
For the last two statements, we use the fact that   vanishes on the subspaces   and   of  . As regards the last term, let  . We have:  . Finally:T˜ξ,t⁎T˜ζ,t(η⊗T)=(cos2⁡(π2t)+sin2⁡(π2t))<ξ,η>(Pη)⊗T+<ξ,η>(Qη)⊗T=<ξ,ζ>η⊗T so  .

We now focus on  . Likewise, we can define   and  .

The element   is given by the Kasparov module
(F(E)⊗iAS⊕F(E)⊗iAS,(π0⊗1S∘iS)⊕(π1⊗1S∘iS),F⊗1S). Then the element   can be represented by the Kasparov module   where   and  .

We also have  .

 Lemma 0.6

Consider the  -subspace   of   (see Proposition 0.1) and let  . Then the representation   in Lemma 0.5 induces a representation  .

Indeed, let   be a generator of the  -algebra S . We first show that   stabilizes  .

Let  .

We have   where
L=∑i=1nλiτ1(Tξi)+∑i=1mμi(τ1(Tζi))⁎M=∑i=1nλiτ0(Tξi)+∑i=1mμi(τ0(Tζi))⁎N=∑i=1nλi(π0(Tξi))⊗1TE+∑i=1mμi(π0(Tξi))⁎)⊗1TE

As in the proof of Lemma 0.5 we only pay attention on the subspaces where terms do not vanish. Let  ,  . Then we have:
(∑i=1nλiτ1(Tξi)+∑i=1mμi(τ1(Tζi))⁎)b=(∑i=1nλiTξi+∑i=1mμiTζi⁎)b∈A⊗iAS⋍S;(∑i=1nλiτ0(Tξi)+∑i=1mμi(τ0(Tζi))⁎)b=(∑i=1nλiξi)⊗b∈E⊗S;(∑i=1nλiτ0(Tξi)+∑i=1mμi(τ0(Tζi))⁎)η⊗b=<∑i=1mμiζi,η>b∈A⊗iAS⋍S. The last term clearly stabilizes  . By linearity,   stabilizes  . As   is continuous on   and  ,   induces a *-morphism   on the involutive algebra generated by g valued in  . We now have to extend   to a morphism on S . We note that   because   is a *-morphism between  -algebras. Then   is continuous and extends to a (unique) morphism on S , still denoted by  . For   or  , we find the same   and   introduced before.

To end the proof, we will show that the family   is a homotopy, and thus  . First we have to show that, for fixed  ,   is continuous. For that, as  , we only have to see it on generators  , which is obvious.

Besides, we need to show that, for   and   fixed, we have:
πtS(b)−π0S(b)∈KS(F(E)⊗iAS). We only need to check it for   generator with  . The projection P , introduced at the beginning, is clearly a compact operator of  , so   is a compact operator of  . We can see that   whereU=∑i=1nλi(πtS(Tξi)−π0S(Tξi))(P⊗1S)V=∑i=1mμi(P⊗1S)(πtS(Tζi⁎)−π0S(Tζi⁎)). Thus we have the relation  .

 Corollary 0.7

We have  . Particularly, we have:
K0(SE)=K0(A)

And thus a different proof of the result of [[4]]:

 Corollary 0.8

Let  . φ is a state of the  -algebra  .

We have  .

Indeed, for   and  , if   and   are the creation operators associated with the vectors   and  , we consider  . It is well known that  , where   (see [[5]]), and that there is an homeomorphism on   onto   that sends the semi-circular measure to the Lebesgue one. That gives rise to an *-isomorphism:
(C([0,1]),φ)⁎r(C([0,1]),φ)⋍(C([−2,2]),ψ)⁎r(C([−2,2]),ψ) so  .

 References

 [1] Pimsner M.V. A class of  -algebras generalizing both Cuntz–Krieger algebras and crossed products by   Free Probability Theory Providence, RI: American Mathematical Society (1997).  189-212 [2] Shlyakhtenko D. Free quasi-free states Pac. J. Math. 1997 ;  177 : 329-368 [cross-ref] [3] Shlyakhtenko D. A-valued semicircular systems J. Funct. Anal. 1999 ;  166 : [4] Voiculescu D. The K-groups of the  -algebra of a semicircular family K-Theory 1993 ;  7 : 5-7 [cross-ref] [5] Voiculescu D., Dykema K., Nica A. Free Random Variables  Providence, RI, USA: American Mathematical Society (1992).