C*-algebras of infinite real rank

Abstract

We introduce the notion of weakly (strongly) infinite real rank for unital $C^{\ast }$-algebras. It is shown that a compact space $X$ is weakly (strongly) infine-dimensional if and only if $C(X)$ has weakly (strongly) infinite real rank. Some other properties of this concept are also investigated. In particular, we show that the group $C^{\ast }$-algebra $C^{\ast }({\mathbb{F}}_{\infty })$ of the free group on countable number of generators has strongly infinite real rank.

9 pages