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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) Westf\"alische Wilhelms-Universit\"at, Mathematisches Institut, Einsteinstrasse 62, D-48149 M\" unster, Germany
(2) Department of Mathematics, Dartmouth College, Hanover, NH 03755--3551, USA

Summary:  We make a detailed study of locally inner actions on $\cs$-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\(X,H^2(G,\T)\)$. 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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