KK-Equivalence for Amalgamated Free Product C*-Algebras

Bass--Serre trees of amalgamated free product C*-algebras

“…Then, the desired sequences will follow from the six-term exact sequences of KK-groups ( [7]) induced from the Toeplitz extension. As a by-product of our approach, we show that ∆T(A, E) is KK-equivalent to A ⊕ D. By the result in [17,15], this implies that the KK-class of ∆T(A, E) is independent of the choice of conditional expectations.…”

“…Since φ Z (P 1 )S = S(σ 1 (P 1 ) ⊗ 1) holds, the triplet (Z ⊕ (X ⊗ D A), φ Z ⊕ (σ 1 ⊗ 1), 0 S S * 0 ) is a ∆T(A, E)-A Kasparov bimodule and defines an element α ∈ KK(∆T(A, E), A). Since φ⊗ ∆T(A,E) α is implemented by the A-A Kasparov bimodule (Z ⊕ (X ⊗ D A), φ Z ⊕ (φ X ⊗ 1), 0 S S * 0 ), we have φ ⊗ ∆T(A,E) α = id A ∈ KK(A, A) by [17,Theorem 3.4] (see [14] for the degenerate case). Also, it follows from φ Z (P 1 ) = S(σ 1 (P 1 ) ⊗ 1)S * that ρ 1 ⊗ ∆T(A,E) α = 0.…”

“…The reduced amalgamated free product of C * -algebras was introduced by Voiculescu in [38]. Since then, various aspects of the construction, for example, simplicity, stable rank [8,10], approximation properties [9,11,29,25], and K-theory [17,14] have been studied. In this paper, we are interested in a new aspect of the construction, namely, an analogue of the boundary actions arising from the Bass-Serre trees of amalgamated free product groups.…”

“…The C * -algebra ∆T(A, E) is generated by a copy of A and two projections, and satisfies that when (A, E) comes from the reduced group C * -algebra C * red (Γ) of Γ = Γ 1 * Λ Γ 2 , one has (C * red (Γ) ⊂ ∆T(C * red (Γ), E)) ∼ = (C * red (Γ) ⊂ C(∆T) ⋊ red Γ). Our construction is inspired by the recent works [17,14] on the K-theory of amalgamated free products. In these works, the KK-equivalence between any reduced amalgamated free product and the corresponding full one is proved.…”

“…Also, we note that in our proof of the exact sequences of KK-groups for (A, E), the semisplit Toeplitz extension of the Cuntz-Pimser algebra identified with ∆T(A, E) plays a key role. In the course of our proof, we will show that the embedding A ֒→ ∆T(A, E) is right invertible in KK-theory using the description of id ∈ KK(A, A) from [17,14]. Then, the desired sequences will follow from the six-term exact sequences of KK-groups ( [7]) induced from the Toeplitz extension.…”