Journal of Operator Theory

Volume 69, Issue 1, Winter 2013 pp. 287-296.

Nuclear and type I

$C^*$ -crossed products

Authors: Raluca Dumitru (1) and Costel Peligrad (2)
Author institution: (1) Department of Mathematics and Statistics, University of North Florida, Jacksonville, FL 32224, U.S.A.
(2) Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221, U.S.A.

Summary: We prove that a

$C^*$ -crossed product

$A\times_{\alpha}G$ by a locally compact group

$G$ is nuclear (respectively type I or liminal) if and only if certain hereditary

$C^*$ -subalgebras,

$S_{\pi}$ ,

$\mathcal{I}_{\pi}\subset A\times_{\alpha}G$

$\pi\in\widehat{K}$ , are nuclear (respectively type I or liminal). Analog characterizations are proved for

$C^*$ -crossed products by compact quantum groups. These subalgebras 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

$A\otimes B(H_{\pi })^{\alpha\otimes \mathrm{ad}\pi}$ where

$H_{\pi}$ is the Hilbert space of the representation

$\pi$ .

DOI: http://dx.doi.org/10.7900/jot.2010dec08.1907
Keywords:

$C^*$ -algebras, crossed products, quantum groups

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