Locally inner actions on C-0(X)-algebras

We make a detailed study of locally inner actions on C*-algebras whose prim itive ideal spaces have locally compact Hausdorff complete regularizations. We suppose that G has a representation group and compactly generated abeli anization G(ab) Then, if A is stable and if the complete regularization of Prim(A) is X, me show that the collection of exterior equivalence classes o f locally inner actions of G on A is parametrized by the group EG(X) of ext erior equivalence classes of C-o(X)-actions of G on C-o(X, K). Furthermore, we exhibit a group isomorphism of epsilon (G)(X) with the direct sum H-1(X , (G) over cap (ab))+ 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.