Real rank and squaring mapping for unital C*-algebras

A. Chigogidze, A. Karasev, M. Rordam

英文摘要：

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

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