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

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 $\dim(X)=0$.