The structure of the Brauer group and crossed products of \(C_0(X)\)-linear group actions on \(C_0(X,\mathcal K)\) (Q2716157)

The paper provides extensions and analogues of the investigations by \textit{S. Echterhoff} and \textit{D. P. Williams} [``Locally inner actions on \(C_0(X)\)-algebras'', Preprint 1997; to appear in J. Operator Theory and J. Funct. Anal. 158, No. 1, 113-151 (1998; Zbl 0909.46055)]. The equivariant Brauer group \(B_G(X)\) [see J. Funct. Anal. 146, No. 1, 151-184 (1997; Zbl 0873.22003)] is described in terms combining the Moore group cohomology and Čech cohomology. If a group \(G\) acts trivially on a space \(X\) then (both \(G\) and \(X\) being separable and locally compact), \(\text{Br}_G(X)=\widetilde H_3(X,{\mathbb Z})\oplus{\mathcal E}_G(X)\), and one studies the subgroup \({\mathcal E}_G(X)\) consisting of all exterior equivalence classes of \(C_0(X)\)-linear actions of \(G\) on \(C_0(X,{\mathcal K})\), where \(\mathcal K\) is the algebra of compact operators on a separable Hilbert space. This task is accomplished here for a wide class of groups \(G\) having a splitting central extension [in the sense of \textit{Calvin C. Moore}, Trans. Am. Math. Soc. 113, 40-63, 64-86 (1964; Zbl 0131.26902)] \(1\to Z\to H\to G\to 1\) with compactly generated abelianization of \(H\). The results extend to give classification of \textit{locally inner} actions \(\alpha:G\to \mathop{\text{ Aut}}(A)\) on (stable) continuous trace \(C^*\)-algebras \(A\) with spectrum \(X\) and even on stable \({\mathcal CR}(X)\)-algebras considered by Echterhoff and Williams. Moreover, the bundle structure of \(C_0(X,{\mathcal K})\rtimes_\alpha G\) is characterized in terms of the topological data associated to the given action \(\alpha\) of \(G\) and the bundle structure of the group \(C^*\)-algebra of \(H\).