The radius of comparison of $C (X)$

TL;DR

This paper investigates the relationship between the radius of comparison of the C*-algebra of continuous functions on a compact Hausdorff space and the space's covering dimension, providing improved bounds and examples of spaces with large or infinite radius of comparison.

Contribution

It establishes a new lower bound for the radius of comparison of C(X) in terms of the covering dimension, refining previous results and illustrating cases with large or infinite radius of comparison.

Findings

Lower bound rc(C(X)) ; rac{ ext{dim}(X) - 7}{2}

Improved upon Elliott and Niu's estimate for metrizable spaces

Existence of spaces with infinite or arbitrarily large finite radius of comparison

Abstract

Let X be a compact Hausdorff space. Then the radius of comparison rc ( C (X)) is related to the covering dimension dim (X) by rc ( C (X)) \geq [ dim (X) - 7 ] / 2. Except for the additive constant, this improves a result of Elliott and Niu, who proved that if X is metrizable then rc (C (X)) \geq [ dim_{\mathbb{Q}} (X) - 4 ] / 2. There are compact metric spaces X for which the estimate of Elliott and Niu gives no information, but for which rc ( C (X)) is infinite or has arbitrarily large finite values.