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On the Radius of Comparison of a Commutative C*algebra


Published:20121023
Printed: Dec 2013
George A. Elliott,
Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4
Zhuang Niu,
Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, Newfoundland and Labrador A1C 5S7
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Abstract
Let $X$ be a compact metric space. A lower bound for the radius of
comparison of the C*algebra $\operatorname{C}(X)$ is given in terms of
$\operatorname{dim}_{\mathbb{Q}} X$, where $\operatorname{dim}_{\mathbb{Q}} X $ is
the cohomological dimension with rational coefficients. If
$\operatorname{dim}_{\mathbb{Q}} X =\operatorname{dim} X=d$, then the
radius of comparison of the C*algebra $\operatorname{C}(X)$ is $\max\{0, (d1)/21\}$ if $d$ is odd, and must be either $d/21$ or $d/22$ if $d$ is even (the possibility of $d/21$ does occur, but we do not know if the possibility of $d/22$ also can occur).