Twisted crossed products by coactions

Abstract

We consider coactions of a locally compact group G on a C*-algebra A, and the associated crossed product C*-algebra A x G. Given a normal subgroup N of G, we seek to decompose A x G as an iterated crossed product (A x G/N) x N, and introduce notions of twisted coaction and twisted crossed product which make this possible. We then prove a duality theorem for these twisted crossed products, and discuss how our results might be used, especially when N is abelian.

Induced 𝐶*-algebras and Landstad duality for twisted coactions

“…Hence, the composite map ^'oOofcj gives an isomorphism of B to C. We show that it conjugates the twisted coactions (8, j) and (e , I) : we have [PR,Section 5(b)] whether every twisted cocrossed product B xG/N G is isomorphic to an ordinary cocrossed product B x N when G splits as a semidirect product over N. Proposition 6.3. Let (B, G, G/N, S, j) be a nondegenerate twisted cosystem, and suppose that there is a continuous section a : G/N -* G. Then there is a coaction e of N on B such that 3 is exterior equivalent to (i ® C) o e .…”

“…Therefore, Katayama's duality theorem applies, and (B x G) x G is isomorphic to B ®Jf . We conclude that (B xG/N G) x N is strongly Morita equivalent to B ®Jf, hence to B. G We remark that nondegeneracy should be added as a hypothesis in [PR,Theorem 4.1], since its proof appeals to Mansfield's imprimitivity theorem [Man,Theorem 28], which uses nondegeneracy. On the other hand, it is tempting to conjecture that coactions twisted over G/N are automatically nondegenerate, since they should behave like coactions of the amenable group N (see [PR, Section 5(b)]).…”

“…For (iii), we need only observe that p\v * (re oj) commutes with both re and p. The second follows by a calculation using Corollary 2.5, while for the first, abelian and (A, G) is a system which is twisted in the sense of [Gre] over a closed subgroup H, then the dual system (A x G, G) is conjugate to the system (Ind^±^4 xH G, G) induced from the twisted crossed product A »H G. By [PR,Remark 2.3] the twisted system (A, G, H) corresponds to a twisted cosystem (A, G, G/IIa-), and we have (Ax-G, G) = (AxG,G), (AxhG,H±) = (Akg/h±G,H±), so Olesen and Pedersen's result follows immediately from Theorem 4.4 above. Theorem 4.4 has a lot of interesting consequences.…”

“…In particular, when G is abelian, a coaction 3 of G has a twist relative to G/N if and only if the corresponding action of G has a Green twisting map on Nx (cf. [PR,2.3]). …”

“…The most interesting of these-and, in our view, the most surprising-is that, if G is trivial as a principal ./V-bundle, then every twisted cocrossed product B xG¡N G is isomorphic to a cocrossed product B xe N by an (untwisted) coaction of N. This is analogous to the corresponding property for crossed products by actions of semidirect products (cf. [PR,Section 5(b)]); the surprise is that we only need a topological splitting, rather than an algebraic one.…”