Nuclear and type I crossed products of C*-algebras by group and compact quantum group actions

Abstract

If A is a C*-algebra, G a locally compact group, K{\subset}G a compact subgroup and {\alpha}:G{\to}Aut(A) a continuous homomorphism, let Ax_{{\alpha}}G denote the crossed product. In this paper we prove that Ax_{{\alpha}}G is nuclear (respectively type I or liminal) if and only if certain hereditary C*-subalgebras, S_{{\pi}}, I_{{\pi}}{\subset}Ax_{{\alpha}}G {\pi}{\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_{{\pi}} are stably isomorphic with the fixed point algebras A{\otimes}B(H_{{\pi}})^{{\alpha}{\otimes}ad{\pi}} where H_{{\pi}} is the Hilbert space of the representation {\pi}.