Tracial oscillation zero and Z-stability

Abstract

Let A be a (not necessarily unital) separable simple amenable C * -algebra whose traces have countable extremal boundary. We show that A is Z-stable if and only if A has strict comparison for positive elements. In the case that A is a unital separable simple amenable C * -algebra whose tracial state space has countable-dimensional extremal boundary. We show that A is Z-stable if and only if A has strict comparison for positive elements and stable rank one. PreliminaryDefinition 2.1. Let A be a C * -algebra and S ⊂ A a subset of A.

Tracial approximation and ${\cal Z}$-stability

“…Then ∂ e (T (E)) = ∪ ∞ n=1 X n . It is σ-compact and countable-dimensional (see [36,Example 2.13]). Put ∆ = T (E).…”

“…Moreover, one would also like to consider simple C * -algebras which are not stably isomorphic to unital ones (the case that A is stably projectionless). This paper makes a step in all three directions (see also [36]).…”

“…Then, for any n ∈ N, there exists a unital homomorphism ψ :M n → p(l ∞ (A)/I ̟ )p.Proof. Since p is a projection, by[36, Proposition 2.22], we may assume that {a n } is a permanent projection lifting of p. It follows from [36, Proposition 2.22 (iii)](see also the claim after (e.2.44) in the proof of[36, Proposition 2.22]) that lim k→∞ ω(a k ) = 0 (seeDefinition 2.10). By[20, Lemma 8.4], for each n ∈ N, there is, for each k ∈ N, an order zero c.p.c.…”

“…The following is also taken from [36] which allows us to add maps with property (Os). Lemma 8.5 ([36, Lemma 4.2]).…”

Hereditary uniform property $Γ$

“…We write P i,l = {p [14] and Proposition 2.21 of [23]). Therefore, by (1) and (2) of Proposition 6.2 of [14] (see also Proposition 2.21 of [23]), we may assume that lim k→̟ sup{τ (p Since τ ((p…”