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Project

Equivariant Z-stability for actions of amenable groups

The main goal of this PhD was to investigate which actions of countable amenable groups on nuclear, simple, separable, Jiang-Su stable C*-algebras are equivariantly Jiang-Su stable. In summary, we obtained two types of positive results regarding this problem:

For unital separable, simple, nuclear, Jiang-Su stable C*-algebras of which the tracial state simplex is sufficiently well-behaved, namely a Bauer simplex with finite-dimensional boundary, we showed that every automorphism is equivariantly Jiang-Su stable when viewed as an action of the integers. We also obtained partial results in this direction for actions of groups with local polynomial growth on such C*-algebras.

In the general case, we proved the equivalence of equivariant Jiang-Su stability and another abstract property, called uniform equivariant property Gamma. As a byproduct of our methods, we showed that actions with uniform equivariant property Gamma satisfy what we call a tracial local-to-global principle. In short, this allows one to derive certain behavior of the C*-dynamical system with respect to the uniform tracial 2-norm from the behavior of the system with respect to the 2-norms arising from the individual tracial states.

Date:29 Aug 2019 →  19 Nov 2023
Keywords:C*-dynamical systems, actions of amenable groups, classification
Disciplines:Functional analysis
Project type:PhD project