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

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$.