Minicourse: An invitation to mean dimension of a dynamical system and the radius of comparison of its crossed product, III

Abstract: The purpose of this minicourse is to explain the background (including the terms below) and some progress towards the following conjecture, relating topological dynamics to the structure of the crossed product $C^*$-algebra.

Let $G$ be a countable amenable group, let $X$ be a compact metrizable space, and let $T$ be an action of $G$ on $X$. The mean dimension $mdim ~(T)$ is a purely dynamical invariant, designed so that the mean dimension of the shift on $([0, 1]^d)^G$ is equal to $d$. The radius of comparison $rc ~(A)$ of a unital $C^*$-algebra $A$ is a numerical measure of failure of comparison in the Cuntz semigroup of $A$, a generalization of unstable K-theory. It was introduced to distinguish $C^*$-algebras having no connection with dynamics. The conjecture asserts that if $T$ is free and minimal, then $rc ~(C^* (G, X, T)) = \frac{1}{2} ~mdim ~(T)$. The inequality $rc ~(C^* (G, X, T)) \leq \frac{1}{2} ~mdim ~(T)$ is known for $G = {\mathbb{Z}}^n$, and progress towards the inequality $rc ~(C^* (G, X, T)) \geq \frac{1}{2} ~mdim ~(T)$ has been made for the known classes of examples of free minimal actions with nonzero mean dimension, for any countable amenable group $G$. The emphasis will be on the inequality $rc ~(C^* (G, X, T)) \geq \frac{1}{2} ~mdim ~(T)$; the results there are joint work with Ilan Hirshberg.

Lecture 3.

In this lecture, we state some known results towards the conjecture $rc ~(C^* (G, X, T)) = \frac{1}{2} ~mdim ~(T)$, and say something about the ideas which go into the results towards the inequality $rc ~(C^* (G, X, T)) \geq \frac{1}{2} ~mdim ~(T)$.

geometric topologynumber theoryoperator algebrasrepresentation theory

Audience: researchers in the topic

( slides | video )

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Noncommutative geometry in NYC

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