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If A is a C*-algebra, G a locally compact group, K{\subset}G a compactsubgroup and {\alpha}:G{\to}Aut(A) a continuous homomorphism, letAx_{{\alpha}}G denote the crossed product. In this paper we prove thatAx_{{\alpha}}G is nuclear (respectively type I or liminal) if and only ifcertain hereditary C*-subalgebras, S_{{\pi}}, I_{{\pi}}{\subset}Ax_{{\alpha}}G{\pi}{\in}K, are nuclear (respectively type I or liminal). These algebras arethe analogs of the algebras of spherical functions considered by R. Godementfor groups with large compact subgroups. If K=G is a compact group or a compactquantum group, the algebras S_{{\pi}} are stably isomorphic with the fixedpoint algebras A{\otimes}B(H_{{\pi}})^{{\alpha}{\otimes}ad{\pi}} whereH_{{\pi}} is the Hilbert space of the representation {\pi}.

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