Classification of finite simple amenable ${\cal Z}$-stable $C^*$-algebras

TL;DR

This paper proves a classification theorem for a broad class of unital simple separable amenable ${ m Z}$-stable $C^*$-algebras using the Elliott invariant, covering all such algebras with certain properties.

Contribution

It establishes a comprehensive classification result for unital simple separable amenable ${ m Z}$-stable $C^*$-algebras, including those satisfying the UCT with finite rational tracial rank.

Findings

Classification by Elliott invariant for this class.

Includes all unital simple separable amenable ${ m Z}$-stable $C^*$-algebras with UCT and finite rational tracial rank.

Exhausts all possible Elliott invariants for these algebras.

Abstract

We present a classification theorem for a class of unital simple separable amenable ${\cal Z}$-stable $C^*$-algebras by the Elliott invariant. This class of simple $C^*$-algebras exhausts all possible Elliott invariant for unital stably finite simple separable amenable ${\cal Z}$-stable $C^*$-algebras. Moreover, it contains all unital simple separable amenable $C^*$-alegbras which satisfy the UCT and have finite rational tracial rank.