Real rank and squaring mapping for unital C*-algebras

Abstract

It is proved that if X is a compact Hausdorff space of Lebesgue dimension $\dim(X)$, then the squaring mapping $\alpha_{m} \colon (C(X)_{\mathrm{sa}})^{m} \to C(X)_{+}$, defined by $\alpha_{m}(f_{1},..., f_{m}) = \sum_{i=1}^{m} f_{i}^{2}$, 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 self-adjoint 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 $\dim(X)=0$.