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


Authors: A. Chigogidze, A. Karasev and M. Rørdam
Journal: Proc. Amer. Math. Soc. 132 (2004), 783-788
MSC (2000): Primary 46L05; Secondary 46L85, 54F45
DOI: https://doi.org/10.1090/S0002-9939-03-07102-8
Published electronically: August 19, 2003
MathSciNet review: 2019956
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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 $\dim(X)=0$.


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Additional Information

A. Chigogidze
Affiliation: Department of Mathematics and Statistics, University of Saskatchewan, McLean Hall, 106 Wiggins Road, Saskatoon, SK, S7N 5E6, Canada
Email: chigogid@math.usask.ca

A. Karasev
Affiliation: Department of Mathematics and Statistics, University of Saskatchewan, McLean Hall, 106 Wiggins Road, Saskatoon, SK, S7N 5E6, Canada
Email: karasev@math.usask.ca

M. Rørdam
Affiliation: Department of Mathematics, University of Southern Denmark, Campusvej 55, 5230 Odense M, Denmark
Email: mikael@imada.sdu.dk

DOI: https://doi.org/10.1090/S0002-9939-03-07102-8
Keywords: Real rank, bounded rank, Lebesgue dimension
Received by editor(s): February 15, 2002
Received by editor(s) in revised form: October 28, 2002
Published electronically: August 19, 2003
Additional Notes: The first named author was partially supported by an NSERC research grant
Communicated by: David R. Larson
Article copyright: © Copyright 2003 American Mathematical Society

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