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 $\text{Prim}(A)$ is $X$, we show that the collection of exterior equivalence classes of locally inner actions of $G$ on $A$ is parameterized by the group ${\text{E}}_G(X)$ of exterior equivalence classes of $C_0(X)-actionsof$G$on$C_0(X,\K)$. Furthermore, we exhibit a group isomorphism of ${\text{E}}_G(X)$ 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 ${\text{Br}}_G(X)$ for $G$ acting trivially on $X$.