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

*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.

*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:

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

*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 .

*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 .

*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 .

*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 .

*We have* *. Particularly, we have:*

K0(SE)=K0(A)

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

*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 .