# The mean dimension of a homeomorphism and the radius of comparison of its C*-algebra

### N. Christopher Phillips ( University of Oregon)

*Tuesday, May 31, 2016,
14:30 – 15:30*,
**Math -101**

**Abstract:**

We describe a striking conjectured relation
between ``dimensions’’ in topological dynamics and C*-algebras.
(No previous knowledge
of C*-algebras or dimension theory will be assumed.)
Let $X$ be a compact metric space,
and let $h \colon X \to X$ be a minimal homeomorphism
(no nontrivial invariant closed subsets).
The *mean dimension*
${\mathit{mdim}} (h)$ of $h$ is a dynamical invariant,
which I will describe in the talk,
and which was invented
for purposes having nothing to do with C*-algebras.
One can also form a crossed product C*-algebra $C^* ({\mathbb{Z}}, X, h)$.
It is simple and unital,
and there is an explicit description
in terms of operators on Hilbert space,
which I will give in the talk.
The *radius of comparison*
${\mathit{rc}} (A)$ of a simple unital C*-algebra $A$
is an invariant introduced for reasons having nothing to do
with dynamics;
I will give the motivation for its definition in the talk
(but not the definition itself).
It has been conjectured,
originally on very thin evidence,
that the radius of comparison of $C^*({\mathbb{Z}},X,h)$
is equal to half the
mean dimension of $h$
for any minimal homeomorphism $h$.

In this talk, I will give elementary introductions to mean dimension, the crossed product construction, and the ideas behind the radius of comparison. I will then describe the motivation for the conjecture and some partial results towards it.