Rocky Mountain Journal of Mathematics

Cuntz-Pimsner algebras of group representations

Valentin Deaconu

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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.

Article information

Rocky Mountain J. Math., Volume 48, Number 6 (2018), 1829-1840.

First available in Project Euclid: 24 November 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46L05: General theory of $C^*$-algebras

$C^*$-correspondence group representation graph algebra Cuntz-Pimsner algebra


Deaconu, Valentin. Cuntz-Pimsner algebras of group representations. Rocky Mountain J. Math. 48 (2018), no. 6, 1829--1840. doi:10.1216/RMJ-2018-48-6-1829.

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