Tracial oscillation zero and stable rank one

Abstract

Let A be a separable (not necessarily unital) simple C * -algebra with strict comparison. We show that if A has tracial approximate oscillation zero then A has stable rank one and the canonical map Γ from the Cuntz semigroup of A to the corresponding affine function space is surjective. The converse also holds. As a by-product, we find that a separable simple C * -algebra which has almost stable rank one must have stable rank one, provided it has strict comparison and the canonical map Γ is surjective.

Tracial oscillation zero and Z-stability

“…Roughly speaking, a separable simple C * -algebra has tracial approximate oscillation zero, if every positive element in A can be approximated (tracially) by those elements whose tracial oscillations tend to zero. This notion will be discussed in detail in [24]. Among other things, we show that if A is a (not necessarily unital) separable simple C * -algebra which has strict comparison, then A has tracial approximate oscillation zero if and only if A has stable rank one and the canonical map Γ is surjective.…”

“…If A also has strict comparison, then A has continuous scale if and only if ω(e) = 0 (see Proposition 5.4 and Theorem 5.3 of [20]). Definition 2.18 (Definition 4.7 of [24]). Let A be a σ-unital compact C * -algebra with T (A) = ∅.…”

“…We assume that all 2-quasitraces of A are traces. Let a ∈ Ped(A ⊗ K) + and let Π : (see Proposition 4.8 of [24]). We say that A has T-tracial approximate oscillation zero, if Ω T (a) = 0 for all a ∈ Ped(A⊗K) + (we still assume that A is compact).…”

“…Suppose that Γ is surjective. Then Γ| Cu(A) + is surjective as well (see Theorem 7.12 [24], for example). Definition 2.9.…”

“…→ 0. This is also equivalent to saying that a n ∈ N cu (A) for m(n) > n (with a little work-see 2.20 of [24], for example). Definition 2.11.…”