Locally Inner Actions on C 0 (X) -Algebras
Abstract
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)−actionsof
G
on
C_0(X,\K)
.Furthermore,weexhibitagroupisomorphismof
\E_G(X)
withthedirectsum
H^1(X,\sheaf \hat{G_{ab}}) \oplus C(X,H^2(G,\T))
.Asaconsequence,wecancomputetheequivariantBrauergroup
\Br_G(X)
for
G
actingtriviallyon
X$.