Tracial approximation in simple -algebras

Abstract

We revisit the notion of tracial approximation for unital simple C *-algebras. We show that a unital simple separable infinite dimensional C *-algebra A is asymptotically tracially in the class of C *-algebras with finite nuclear dimension if and only if A is asymptotically tracially in the class of nuclear Z-stable C *-algebras.

Non-amenable simple C*-algebras with tracial approximation

“…Remark 5.8. In [17], we show that a separable simple C * -algebra which is tracially approximately divisible has strict comparison for positive elements. So there is a redundancy in the assumption of Theorem 5.7.…”

“…But g(1 − e)x ∈ A. In the case that A is unital, the condition (1 − e)x ≥ x − ε for all x ∈ F in (2) of the definition is redundant for most cases (we leave the discussion to [17]).…”

Tracial approximate divisibility and stable rank one

“…For any η > 0, since A is simple, infinite-dimensional, and non-elementary, Her A (e) is also simple, infinite-dimensional, and non-elementary. It follows from [15,Lemma 4.3] (see also [23,Lemma 2.4]) that there is a η ∈ Her A (e) + \ {0} such that d τ (a η ) < η for all τ ∈ T e . There exists an integer…”

“…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).…”

“…. Suppose now that A is non-elementary, separable and simple, by [15,Lemma 4.3], there exists a sequence {s n } ⊂ A + with s n = 1 such that, for any a ∈ A + \ {0}, there exists n 0 ≥ 1 such that s n a for all n ≥ n 0 . In other words, {s n } ∈ N cu (A).…”