Tracial approximate divisibility and stable rank one

Abstract

In this paper, we show that every separable simple tracially approximately divisible C *algebra has strict comparison, and, it is either purely infinite or has stable rank one. As a consequence, we show that every (non-unital) finite simple Z-stable C * -algebra has stable rank one.

Tracial oscillation zero and stable rank one

“…The following is known. The proof of this is exactly the same as that of the case T (A) = QT (A) (see 5.1, 5.2, 5.3 and 5.4 of [13] for details, and also see the remark after Definition 6.3 of [17]).…”

“…Therefore, for any hereditary C * -subalgebra A 1 ⊂ A, any a ∈ A 1 , any ε > 0, there are nilpotents x, y ∈ A 1 such that a − xy < ε. Together with the facts that A is projectionless and has continuous scale, we can apply [17,Theorem 6.4] and conclude that A has stable rank one. We learned that the following is also obtained by Winter and Geffen.…”

“…Denote by N cu (A) (or just N cu ) the set of all Cuntz-null sequences in l ∞ (A). It follows from Proposition 3.5 of [17] that…”

Tracial oscillation zero and Z-stability

“…Denote by N cu := N cu (A) the subset of l ∞ (A) consisting of Cuntz-null sequences. It is proved in [23] that, if A is a nonelementary simple C * -algebra, N cu (A) is a (closed two-sided) ideal of l ∞ (A) containing c 0 (A) (see Proposition 3.5 of [23]). Let e ∈ A 1 + be a strictly positive element and e n = f 1/2n (e), n ∈ N. Recall that (see Definition 2.5 of [35]) A is said to have continuous scale if and only if for any m(n) > n, a n = (e m(n) − e n ) c.…”

“…It is shown (Proposition 3.8 of [23]) that, with additional assumption that A has strict comparison, N cu (A) = I T (A) w ,N . Definition 2.12.…”