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Cuntz-Pimsner algebras of group representations. (English) Zbl 1409.46031
Summary: Given a locally compact group \(G\) and a unitary representation \(\rho :G\rightarrow U({\mathcal H})\) on a Hilbert space \(\mathcal{H}\), we construct a \(C^\ast\)-correspondence \({\mathcal E}(\rho)={\mathcal H}\otimes_{\mathbb C}C^\ast(G)\) over \(C^\ast(G)\) 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 \(C^\ast\)-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 \(C^\ast(G)\). 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 \(C^\ast\)-correspondence.
MSC:
46L05 General theory of \(C^*\)-algebras
22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
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