On locally AH algebras

Abstract

We show that every unital amenable separable simple C * -algebra with finite tracial rank which satisfies the UCT has in fact tracial rank at most one. We also show that unital separable simple C * -algebras which are "tracially" locally AH with slow dimension growth are Z-stable. As a consequence, unital separable simple C * -algebras which are locally AH with no dimension growth are isomorphic to a unital simple AH-algebra with no dimension growth.

High-dimensionalZ-stable AH algebras

“…This result clarifies the role played by simplicity in classification results such as [4,6,9]. It also provides the first example of a Z-stable approximately subhomogeneous algebra A which cannot be approximated by subhomogeneous algebras with bounded decomposition rank; it is expected (and largely entailed by classification conjectures) that this phenomenon cannot occur in the simple case.…”

“…Classification arguments, on the one hand, show that for a simple algebra in AC, if it is an inductive limit of building blocks (in C) with bounded topological dimension (or even "slow dimension growth"), or if it is Z-stable, then it is an inductive limit of algebras in C with topological dimension at most three [4,6,9]. (Note that one can show, without classification, that slow dimension growth implies Z-stability; see [11,12,16], so Zstability should be viewed as the weakest of these hypotheses.)…”

Minimal Dynamical Systems on Connected Odd Dimensional Spaces

“…Therefore (e 3.16) holds. It follows from Theorem 1.1 of [13] that A x is a unital simple AH-algebra with no dimension growth. Proposition 3.3.…”

Approximated by finite-dimensional homomorphisms into simple $C^*$-algebras with tracial rank one

“…Recently, H. Lin proved that if a unital simple C*-algebra A with T R(A) ≤ k satisfies the UCT, then A actually has tracial rank no more that one (see [Lin11]). Hence we focus on A with T R(A) ≤ 1.…”