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

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

1997-06-17

arxiv.org/abs/funct-an/9706002v1

Mathematics

Operator Algebras

33 pages AMS-LaTeX

Scientific paper

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 $X$.

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