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

TL;DR

This paper studies inductive limit actions of compact groups on AF algebras, showing that non-inner actions are not classifiable by standard equivariant K-theory and providing a complete classification via twisted equivariant K-theory.

Contribution

It introduces a classification framework for non-inner inductive limit actions on AF algebras using twisted equivariant K-theory, extending prior results.

Findings

Non-inner actions are not classifiable by equivariant K-theory.

Complete classification achieved using twisted equivariant K-theory.

Inner actions are classified by existing equivariant K-theory.

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 $\alpha_n\colon G\curvearrowright A_n$ 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.