# 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 $\pi$.

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

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