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# On the Radius of Comparison of a Commutative C*-algebra

Published:2012-10-23
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 $\max\{0, (d-1)/2-1\}$ if $d$ is odd, and must be either $d/2-1$ or $d/2-2$ if $d$ is even (the possibility of $d/2-1$ does occur, but we do not know if the possibility of $d/2-2$ also can occur).
 Keywords: Cuntz semigroup, comparison radius, cohomology dimension, covering dimension