Christopher Phillips

University of Oregon, USA

The Cuntz semigroup and radius of comparison of C*-algebras from dynamics

This is joint work with Ilan Hirshberg.

The radius of comparison  rc (A)  of a C*-algebra  A is a measure of the failure of the order on the Cuntz semigroup, a kind of expanded version of the  K_0-group, to be determined by traces on the algebra. It was introduced by Toms to distinguish counterexamples in the Elliott classification program. Zero radius of comparison is the best case, and roughly corresponds to classifiability of the algebra.

For an action of an amenable group  G  on a compact metric space  X, the mean dimension  mdim (G, X) is a dynamical invariant designed to give the value  d  on the shift on  ([0, 1]^d)^G. Its motivation was unrelated to C*-algebras or their (generalized) K-theory, and the algebras Toms 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 the concepts, there is significant evidence for the conjecture that  rc ( C^* (G, X) ) = (1/2) mdim (G, X) when the action is free and minimal. We will explain the concepts above, with emphasis on the Cuntz semigroup and radius of comparison; no previous knowledge of these, of C*-algebras, or of mean dimension will be assumed. Then we describe some of the evidence. In particular, we give the first general partial results towards the direction  rc ( C^* (G, X) ) \geq (1/2) 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  mdim (G, X) > 0.