Nuclear and type I crossed products of C*-algebras by group and compact quantum group actions
arXiv:1012.6005
Abstract
If A is a C*-algebra, G a locally compact group, K{\subset}G a compact subgroup and α:G{\to}Aut(A) a continuous homomorphism, let Ax_{α}G denote the crossed product. In this paper we prove that Ax_{α}G is nuclear (respectively type I or liminal) if and only if certain hereditary C*-subalgebras, S_{π}, I_{π}{\subset}Ax_{α}G π{\in}K, are nuclear (respectively type I or liminal). These algebras are the analogs of the algebras of spherical functions considered by R. Godement for groups with large compact subgroups. If K=G is a compact group or a compact quantum group, the algebras S_{π} are stably isomorphic with the fixed point algebras A{\otimes}B(H_{π})^{α{\otimes}adπ} where H_{π} is the Hilbert space of the representation π.
8 pages. Submitted