The radius of comparison of C (X) is about half the covering dimension of X

Abstract: Recall that a C*-algebra $A$ has strict comparison of projections if whenever $p$ and $q$ are projections in matrix algebras over $A$, and $\tau (p) < \tau (q)$ for all tracial states $\tau$ on $A$, then $p$ is Murray-von Neumann subequivalent to $q$. In connection with the Elliott classification program, and because many simple C*-algebras have very few projections, this has been extended to comparison of general positive elements. (This will be explained in the talk.) Strict comparison holds for unital stably finite classifiable simple C*-algebras. The radius of comparison ${\mathrm{rc}} (A)$ of a C*-algebra $A$ is a numerical measure of the failure of strict comparison. It is zero if strict comparison holds, and in general is a not so well understood kind of topological dimension.

Let $X$ be a compact metric space. It has been known for some time that ${\mathrm{rc}} (C (X))$ is at most about half the covering dimension of $X$. In 2013, Elliott and Niu proved that ${\mathrm{rc}} (C (X))$ is, up to an additive constant, at least half the rational cohomological dimension of $X$. Recently, we proved that, up to a slightly worse additive constant, ${\mathrm{rc}} (C (X))$ is at least half the covering dimension of $X$, which is sometimes much larger. This shows that ${\mathrm{rc}} (A)$, like stable rank, roughly corresponds to covering dimension, not to rational or integral cohomological dimension, and not to some previously unknown dimension.

geometric topologynumber theoryoperator algebrasrepresentation theory

Audience: researchers in the topic

( paper | slides | video )

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