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

Research paper by **Mikael Rordam**

Indexed on: **30 Aug '04**Published on: **30 Aug '04**Published in: **Mathematics - Operator Algebras**

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#### 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.