A universal multicoefficient theorem for the Kasparov groups

Abstract

Let K(A) denote the sum of all the K-theory groups of a C*-algebra A in all degrees and with all cyclic coefficient groups. The Bockstein operations (which generate a category Λ) act on K(A). We establish a universal coefficient exact sequence 0 → Pext(K * (A), K * (B)) δ − → KK(A, B) Γ − → Hom Λ (K(A), K(B)) → 0. that holds in the same generality as the universal coefficient theorem of Rosenberg and Schochet. There are advantages, in some circumstances, to using Hom Λ (K(A), K(B)) in place of KK(A, B). These advantages derive from the fact that K(A) can be equipped with order and scale structures similar to those on K 0 (A). With this additional structure, the "Λ − module" K(A) becomes a powerful invariant of C*algebras. We show that it is a complete invariant for the class of real-rank-zero AD algebras. The AD algebras are a certain kind of approximately subhomogeneous C *-algebras which may have torsion in K 1 [Ell]. In addition to classifying these algebras, we calculate their automorphism groups up to approximately innerautomorphisms.

On classification of non-unital amenable simple C*-algebras, II

“…We have, by the definition of z 0 , for a ⊗ c ∈ F (see also (12) in the proof of [90]), Ad z ′′ 0 ((ϕ [1,2,4] )(a ⊗ 1 Zp,q ) ⊗ (θ • ̺km • κ 2km ) [3] (c)) (e 14.13) = (ϕ [1,2,4] ⊗ id…”

“…there exists a unital When K * (A) is finitely generated, Hom Λ (K(A), K(B)) is determined by a finitely generated subgroup of K(A) (see [12]). Let P be a finite subset which generates this subgroup.…”

“…Therefore, without loss of generality, we may assume that g i = [v i ], i = 1, 2, ..., k, and m 0 = k. By 2.11 of [12], since K i (A) is finitely generated (i = 0, 1), there exists K ≥ 1 such that…”

“…Z ) • (id A⊗Z ⊗(θ ⊗ γ 2km+1 ) [3,4] )(z 0 ) ∈ (B ⊗ Z ⊗ Z ⊗ Z p,q ) ∼ (as u ′′′ in the proof of Lemma 3.4 of [90]). By the definition of z 0 and recalling the fact that θ(1 C km+1 ) = 1 Z , for all a ⊗ c ∈ F (see [12] of the proof of Lemma 3.4 of [90]), Ad u ′′′ • ϕ [1,2,4] [3,4] [3,4]…”

“…) is finitely generated, there is an integer T (k) ≥ 1 such that T (k)x = 0 for all x ∈ Tor(K i (A k )) (i = 0, 1). It follows from 2.11 of [12] (note that A k satisfies UCT) that Hom…”