Journal of Operator Theory

Volume 45, Issue 1, Winter 2001 pp. 131-160.

Locally inner actions on $C_0(X)$-algebras

Authors: Siegfried Echterhoff (1), and Dana P. Williams (2)
Author institution: (1) Westfälische Wilhelms-Universität, Mathematisches Institut, Einsteinstrasse 62, D-48149 Münster, Germany
(2) Department of Mathematics, Dartmouth College, Hanover, NH 03755-3551, USA

Summary: $\def\gab{G_{\subab}}$ $\def\prima{{\Prim(A)}}$ $\def\E{{\cal E}}$ $\def\K{{\cal K}}$ $\def\cox{{C_0(X)}}$ $\def\coxk{C_0(X,\K)}$ $\def\T{{\bbb T}}$ $\def\br{{{\rm Br}}}$ $\def\shgab{\widehat{\sheaf G}_{\subab}}$ $\def\subab{{\rm ab}}$ $\def\Prim{{\rm Prim}}$ $\def\sheaf#1{{\cal #1}}$ 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 $\gab$. Then, if $A$ is stable and if the complete regularization of $\prima$ is $X$, we show that the collection of exterior equivalence classes of locally inner actions of $G$ on $A$ is parametrized by the group $\E_G(X)$ of exterior equivalence classes of $\cox$-actions of $G$ on $\coxk$. Furthermore, we exhibit a group isomorphism of $\E_G(X)$ with the direct sum $H^1(X,\shgab)\oplus C(X,H^2(G,\mathbb T))$. As a consequence, we can compute the equivariant Brauer group $\br_G(X)$ for $G$ acting trivially on $X$.

Keywords: Crossed products, locally inner action, $C_0(X)$-algebras, exterior equivalence, equivariant Brauer group

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