Classification of certain inductive limit actions of compact groups on AF algebras

Abstract

Let $A=\underrightarrow{\lim }{A}_n$ be an AF algebra, $G$ be a compact group. We consider inductive limit actions of the form $\alpha =\underrightarrow{\lim }{\alpha }_n$, where ${\alpha }_n:G\curvearrowright A_n$ is an action on the finite dimensional C*-algebra $A_n$ which fixes each matrix summand. If each ${\alpha }_n$ is inner, such actions are classified by equivariant K-theory by Handelman and Rossmann. However, if the actions ${\alpha }_n$ are not inner, we show that such actions are not classifiable by equivariant K-theory. We give a complete classification of such actions using twisted equivariant K-theory.

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