Speaker: **N. Christopher Phillips** (Univ. Oregon)

Title: Radius of comparison for C^{*}
crossed products by free minimal actions of amenable groups

Time/Date: 4:45-6:15, *Friday*, July 22, 2016

Room: 118

Abstract:
The von Neumann algebra crossed product for an ergodic probability
measure preserving action of a countable amenable group on a standard
measure space is always well behaved: it is the hyperfinite factor of
type II_{1}. The C^{*} analog is the crossed product
by a free minimal action
of a countable amenable group on a compact metric space. Much remains
unknown, but a conjectural picture is emerging. First, as has been known
for some time, there is no uniqueness statement similar to that for the
hyperfinite II_{1} factor.
The algebras are all nuclear (the C^{*} version of
hyperfiniteness), but even in the best behaved cases, many different
algebras can arise. The best behaved cases are covered by the Elliott
classification program, but there are badly behaved simple separable
nuclear stably finite C^{*}-algebras, and such algebras can arise as
crossed products even for minimal homeomorphisms. The general conjecture
relates the radius of comparison, a measure of how badly behaved the
crossed product is, to a dynamical quantity called the mean dimension.

In this talk, I will describe the basic setup, the radius of comparison, and mean dimension. I will describe what is known so far; even when the group is Z, only partial results towards the conjecture are known.