Applications of model theory to C*-dynamics

Abstract

We initiate the study of compact group actions on C*-algebras from the perspective of model theory, and present several applications to C*-dynamics. Firstly, we prove that the continuous part of the central sequence algebra of a strongly self-absorbing action is indistinguishable from the continuous part of the sequence algebra, and in fact equivariantly isomorphic under the Continuum Hypothesis. As another application, we present a unified approach to several dimensional inequalities in C*-algebras, which is done through the notion of order zero dimension for an (equivariant) *-homomorphism. Finiteness of the order zero dimension implies that the dimension of the target algebra can be bounded by the dimension of the domain. The dimension can be, among others, decomposition rank, nuclear dimension, or Rokhlin dimension. As a consequence, we obtain new inequalities for these quantities. As a third application we obtain the following result: if a C*-algebra $A$ absorbs a strongly self-absorbing C*-algebra $D$, and $\alpha $ is an action of a compact group $G$ on $A$ with finite Rokhlin dimension with commuting towers, then $\alpha $ absorbs any strongly self-absorbing action of $G$ on $% D$. This has a number of interesting consequences, already in the case of the trivial action on $D$. For example, we deduce that $D$-stability passes from $A$ to the crossed product. Additionally, in many cases of interest, our result restricts the possible values of the Rokhlin dimension to $0, 1$ and $\infty$, showing a striking parallel to the behavior of the nuclear dimension for simple C*-algebras. We also show that an action of a finite group with finite Rokhlin dimension with commuting towers automatically has the Rokhlin property if the algebra is UHF-absorbing.