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.
Categories: math.GN math.FA