On Local AH algebras

TL;DR

This paper proves that certain simple, unital, amenable $C^*$-algebras with finite tracial rank have at most tracial rank one, and establishes conditions under which they are ${ m Z}$-stable or isomorphic to AH-algebras.

Contribution

It demonstrates that unital amenable simple $C^*$-algebras with finite tracial rank satisfying the UCT have tracial rank at most one, and characterizes when they are ${ m Z}$-stable or isomorphic to AH-algebras.

Findings

Algebras with finite tracial rank satisfying UCT have tracial rank ≤ 1.

Tracially locally AH algebras with slow dimension growth are ${ m Z}$-stable.

Locally AH algebras with no dimension growth are isomorphic to 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 $C^*$-algebrass which are "tracially" locally AH with slow dimension growth are ${\cal Z}$-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.