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 II₁. 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 II₁ 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.