Twisted crossed products of \(C^*\)-algebras. II
This is a sequel to an earlier paper of the authors [Math. Proc. Camb. Phil. Soc. 106, 293-311 (1989)], in which they discussed twisted actions (\(\alpha\),u) of a locally compact group G on a \(C^*\)-algebra A, and the associated twisted crossed product \(A\times_{\alpha,u}G\). The main theorem there asserted that every \(A\times_{\alpha,u}G\) is stably isomorphic to an ordinary crossed produced (A\(\otimes {\mathcal K}(H))\times_{\beta}G\), and the present paper gives several new applications. In the first, it is shown that if A is the algebra of sections of a \(C^*\)-bundle, and the automorphisms preserve the fibres, then \(A\times_{\alpha,u}G\) is also given by a \(C^*\)-bundle; this generalises recent results of \textit{D. P. Williams} [J. Austral. Math. Soc., to appear] and \textit{M. A. Rieffel} [Math. Ann. 283, 631-643 (1989; Zbl 0646.46063)]. The authors then compute the K-theory of the twisted transformation group algebras \(C_ 0(X)\times_{\tau,\omega}\Gamma\) associated to actions of a discrete subgroup \(\Gamma\) of a solvable Lie group G, in terms of the twisted K-theory of the orbit space (G\(\times X)/\Gamma\) for the diagonal action. Finally, they show that homotopic twisted actions of such a \(\Gamma\) give crossed products with the same K- theory. All these results have special cases which were previously known (e.g. when X is a point, or \(\Gamma ={\mathbb{Z}}^ n)\).
algebra of sections of a \(C^ *\)- bundle
K-theory of the twisted transformation group algebras
scientific article; zbMATH DE number 4117244
scientific article