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

TL;DR

This paper characterizes when crossed products of C*-algebras by group and quantum group actions are nuclear or type I, using hereditary subalgebras related to spherical functions and fixed point algebras.

Contribution

It establishes necessary and sufficient conditions for nuclearity and type I properties of crossed products via hereditary subalgebras, extending classical results to quantum groups.

Findings

Crossed product nuclearity is characterized by hereditary subalgebras.

Type I and liminal properties are similarly characterized.

Hereditary subalgebras relate to fixed point algebras in quantum group cases.

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}.