Real rank and squaring mappings for unital $C^{\ast }$-algebras

Abstract:

It is proved that if $X$ is a compact Hausdorff space of Lebesgue dimension $\dim (X)$, then the squaring mapping $\alpha _{m} \colon \left ( C(X)_{\mathrm {sa}}\right )^{m} \to C(X)_{+}$, defined by $\alpha _{m}(f_{1},\dots ,f_{m}) = \sum _{i=1}^{m} f_{i}^{2}$, is open if and only if $m -1 \ge \dim (X)$. Hence the Lebesgue dimension of $X$ can be detected from openness of the squaring maps $\alpha _m$. In the case $m=1$ it is proved that the map $x \mapsto x^2$, from the selfadjoint elements of a unital $C^{\ast }$-algebra $A$ into its positive elements, is open if and only if $A$ is isomorphic to $C(X)$ for some compact Hausdorff space $X$ with $\dim (X)=0$.