Locally Inner Actions on $C_0(X)$-Algebras

TL;DR

This paper characterizes locally inner actions on certain C*-algebras via topological and cohomological invariants, providing explicit group isomorphisms and computing the equivariant Brauer group for trivial actions.

Contribution

It introduces a parameterization of locally inner actions using cohomological invariants and computes the equivariant Brauer group for trivial group actions on the space.

Findings

Parameterization of locally inner actions by cohomological groups

Explicit isomorphism of action classes with cohomological invariants

Calculation of the equivariant Brauer group for trivial actions

Abstract

We make a detailed study of locally inner actions on C*-algebras whose primitive ideal spaces have locally compact Hausdorff complete regularizations. We suppose that $G$ has a representation group and compactly generated abelianization $G_{ab}$. Then if the complete regularization of $\Prim(A)$ is $X$, we show that the collection of exterior equivalence classes of locally inner actions of $G$ on $A$ is parameterized by the group $\E_G(X)$ of exterior equivalence classes of $C_0(X)-actions of $G$ on $C_0(X,\K)$. Furthermore, we exhibit a group isomorphism of $\E_G(X)$ with the direct sum $H^1(X,\sheaf \hat{G_{ab}}) \oplus C(X,H^2(G,\T))$. As a consequence, we can compute the equivariant Brauer group $\Br_G(X)$ for $G$ acting trivially on $X$.