Classification of simple $C^*$-algebras of tracial topological rank zero

TL;DR

This paper establishes a classification theorem for unital separable simple nuclear $C^*$-algebras with tracial topological rank zero, showing they are isomorphic if their K-theoretic invariants match, under the Universal Coefficient Theorem.

Contribution

It provides a complete classification of a class of $C^*$-algebras using K-theoretic invariants, extending the understanding of their structure.

Findings

Classification based on K-theoretic invariants is complete for these algebras.

Isomorphism is determined by matching $K_0$, $K_1$, and related data.

The result applies to algebras satisfying the Universal Coefficient Theorem.

Abstract

We give a classification theorem for unital separable simple nuclear $C^*$-algebras with tracial topological rank zero which satisfy the Universal Coefficient Theorem. We prove that if $A$ and $B$ are two such $C^*$-algebras and $$ (K_0(A), K_0(A)_+, [1_A], K_1(A)) $$ $$ \cong (K_0(B), K_0(B)_+, [1_B], K_1(B)), $$ then $A\cong B.$