Simple nuclear $C^*$-algebras of tracial topological rank one

TL;DR

This paper provides a classification theorem for unital separable nuclear simple C*-algebras with tracial rank at most one, establishing isomorphism criteria based on K-theory and trace data.

Contribution

It introduces a classification result for a class of nuclear C*-algebras with low tracial rank, extending the understanding of their structure.

Findings

A classification theorem for C*-algebras with TR ≤ 1.

Isomorphism characterized by K-theory and trace invariants.

Applicable to algebras satisfying the universal coefficient theorem.

Abstract

We give a classification theorem for unital separable nuclear simple \CA s with tracial rank no more than one. Let $A$ and $B$ be two unital separable simple nuclear \CA s with $TR(A), TR(B)\le 1$ which satisfy the universal coefficient theorem. We show that $A\cong B$ if and only if $$ (K_0(A), K_0(A)_+, [1_A], K_1(A), T(A)) \cong (K_0(B), K_0(B)_+, [1_B], K_1(B), T(B)). $$