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 ${\alpha }_m:(C(X{)}_{\mathrm{sa}}{)}^m\to C(X{)}_{+}$, 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 $\dim (X)=0$.