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2018 Cuntz-Pimsner algebras of group representations
Valentin Deaconu
Rocky Mountain J. Math. 48(6): 1829-1840 (2018). DOI: 10.1216/RMJ-2018-48-6-1829

Abstract

Given a locally compact group $G$ and a unitary representation $\rho :G\to U({\mathcal H})$ on a Hilbert space $\mathcal {H}$, we construct a $C^*$-correspondence ${\mathcal E}(\rho )={\mathcal H}\otimes _{\mathbb C} C^*(G)$ over $C^*(G)$ and study the Cuntz-Pimsner algebra ${\mathcal O}_{{\mathcal E}(\rho )}$. We prove that, for $G$ compact, ${\mathcal O}_{{\mathcal E}(\rho )}$ is strongly Morita equivalent to a graph $C^*$-algebra. If $\lambda $ is the left regular representation of an infinite, discrete and amenable group $G$, we show that ${\mathcal O}_{{\mathcal E}(\lambda )}$ is simple and purely infinite, with the same $K$-theory as $C^*(G)$. If $G$ is compact abelian, any representation decomposes into characters and determines a skew product graph. We illustrate with several examples, and we compare ${\mathcal E}(\rho )$ with the crossed product $C^*$-correspondence.

Citation

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Valentin Deaconu. "Cuntz-Pimsner algebras of group representations." Rocky Mountain J. Math. 48 (6) 1829 - 1840, 2018. https://doi.org/10.1216/RMJ-2018-48-6-1829

Information

Published: 2018
First available in Project Euclid: 24 November 2018

zbMATH: 06987227
MathSciNet: MR3879304
Digital Object Identifier: 10.1216/RMJ-2018-48-6-1829

Subjects:
Primary: 46L05

Rights: Copyright © 2018 Rocky Mountain Mathematics Consortium

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Vol.48 • No. 6 • 2018
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