Real rank and squaring mappings for unital $C^{\ast }$-algebras
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- by A. Chigogidze, A. Karasev and M. Rørdam PDF
- Proc. Amer. Math. Soc. 132 (2004), 783-788 Request permission
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 $\dim (X)=0$.References
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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
- 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
- © Copyright 2003 American Mathematical Society
- 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
- MathSciNet review: 2019956