Functional Analysis and Applications Seminar 

19/03/2010, 15:00 — 16:00 — Room P3.10, Mathematics Building
N. Christopher Phillips, University of Oregon, USA

The structure of C*-algebras of free minimal actions of $Z^d$

Let $X$ be a compact metric space with finite covering dimension, and equipped with a free minimal action of $Z^d.$ When $d=1,$ a fair amount is understood about the transformation group C*-algebra $C^{*}(Z^d,X),$ but for $d>1$ very little is known about it except when $X$ is the Cantor set. We describe how to prove, under an additional technical condition, that $C^{*}(Z^d,X)$ has strict comparison for positive elements. This condition says, roughly speaking, that the order on the Cuntz semigroup of $C^{*}(Z^d,X)$ is determined by the tracial states on $C^{*}(Z^d,X),$ which in this case all come from invariant probability measures on~$X.$ We use this result to deduce the more familiar condition, that the order on projections over $C^{*}(Z^d,X)$ is determined by the tracial states. However, we do not know how to prove this result without using the Cuntz semigroup. The technical condition is satisfied whenever $X$ is a smooth manifold and the action is via diffeomorphisms. In this talk, I will focus on the part of the proof involving the Cuntz semigroup, and how the Cuntz semigroup of a suitable ''large'' subalgebra of $C^{*}(Z^d,X)$ can be used to obtain information about the Cuntz semigroup of $C^{*}(Z^d,X)$ itself.