On tracial $\mathcal Z$-stability of simple non-unital C*-algebras

Abstract

We investigate the notion of tracial Z-stability beyond unital C * -algebras, and we prove that this notion is equivalent to Z-stability in the class of separable simple nuclear C * -algebras.

Tracial approximate divisibility and stable rank one

“…Remark 4.14. We note that the implication from (TAD-2) to the uniform McDuff property is also proved in [8,Proposition 4.6].…”

“…Remark 5.3. It is worth noting that Theorem 5.2 has also been independently proved in [8] with a different point of view. More precisely, [8,Theorem 3.2] shows that (non-unital) σ-unital simple C * -algebras with (TAD-2) have strict comparison.…”

“…Similar variations of tracially approximate divisibility also occurred (see Definition 4.4 below, and also in Definition 10.1 of [11]). A concept with the same nature was also given in [18] which was called tracially Z-absorbing (see also [8] and [1]). As a continuation of [16] (and also of [15]), we first show that these notions of tracially approximate divisibility are all equivalent for (not necessarily unital) non-elementary separable simple C * -algebras (see Theorem 4.11).…”

Tracial oscillation zero and Z-stability

“…Hence (2) follows from a result of Matui-Sato (see, explicitly, Corollary 5.11 and Proposition 5.12 of [32], for example) in the unital case. For non-unital case, we take a detour and use the result in [11]. In this case, since T (A 1 ) is compact, A 1 is uniformly McDuff (see Definition 4.1 of [11], or Definition 4.2 of [10]).…”

“…For non-unital case, we take a detour and use the result in [11]. In this case, since T (A 1 ) is compact, A 1 is uniformly McDuff (see Definition 4.1 of [11], or Definition 4.2 of [10]). Since A 1 has strict comparison, by Proposition 4.4 of [11] (also a version of Matui-Sato's result), we conclude that A 1 ∼ = A 1 ⊗ Z.…”