The stable and the real rank of Z-absorbing C*-algebras

TL;DR

This paper investigates properties of Z-absorbing C*-algebras, establishing conditions under which they are purely infinite, traceless, or have stable and real ranks, advancing classification theory.

Contribution

It provides new characterizations of Z-absorbing C*-algebras related to their rank, infiniteness, and tracelessness, extending understanding of their structure.

Findings

W(A) is weakly unperforated for Z-absorbing algebras

A is purely infinite iff A is traceless if A is exact

A has stable rank one iff A is finite if simple and unital

Abstract

Suppose that A is a C*-algebra for which A is isomorphic to A tensor Z, where Z is the Jiang-Su algebra: a unital, simple, stably finite, separable, nuclear, infinite dimensional C*-algebra with the same Elliott invariant as the complex numbers. We show that: (i) The Cuntz semigroup W(A) of equivalence classes of positive elements in matrix algebras over A is weakly unperforated. (ii) If A is exact, then A is purely infinite if and only if A is traceless. (iii) If A is separable and nuclear, then A is isomorphic to A tensor O_infty if and only if A is traceless. (iv) If A is simple and unital, then the stable rank of A is one if and only if A is finite. We also characterise when A is of real rank zero.