# Colloquium: Tuesday March 21, 2023. Speaker: Christopher Phillips (Oregon). Title: “Relations between dynamics and C*-algebras: Mean dimension and radius of comparison”.

room 614, Science & Education building.

A zoom link for our meetings is:

https://us02web.zoom.us/j/83337601824

Speaker : Christopher Phillips (Oregon)

Date : Tuesday, 21st of March, 2023.

Time : 14:00

Title: Relations between dynamics and C*-algebras: Mean dimension and

Abstract: For an action of an amenable group $G$on a compact metric space~$X$, the mean dimension ${\mathrm{mdim}} (G, X)$

was introduced by Lindenstrauss and Weiss.
It is designed so that the mean dimension of the shift on
$([0, 1]^d)^G$is~$d$. Its motivation was unrelated to C*-algebras.

The radius of comparison ${\mathrm{rc}} (A)$of a C*-algebra~$A$
was introduced by Toms to distinguish counterexamples
in the Elliott classification program.

The algebras he used have nothing to do with dynamics.

A construction called the crossed product associates
a C*-algebra $C^* (G, X)$to a dynamical system $(G, X)$.
Despite the apparent lack of connection between these concepts,
there is significant evidence for the conjecture that
${\mathrm{rc}} ( C^* (G, X) ) = \frac{1}{2} {\mathrm{mdim}} (G, X)$
when the action is free and minimal.
We will explain the concepts above; no previous knowledge
of mean dimension, C*-algebras, or radius of comparison will be assumed.
Then we describe some of the evidence.
In particular, we give the first general partial results
towards the direction
${\mathrm{rc}} ( C^* (G, X) ) \geq \frac{1}{2} {\mathrm{mdim}} (G, X)$.
We don’t get the exact conjectured bound,
but we get nontrivial results for many of the known examples
of free minimal systems with ${\mathrm{mdim}} (G, X) > 0″>.

This is joint work with Ilan Hirshberg.

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