2016/12/28 by Deaconu, Valentin
#FOS: Mathematics #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.1612.08979
Given a locally compact group G and a unitary representation ρ:G→ U(\mathcal H) on a Hilbert space \mathcal H, we construct a C^*-correspondence \mathcal E(ρ)=\mathcal H⊗\mathbb C C^*(G) over C^*(G) and study the Cuntz-Pimsner algebra \mathcal O_\mathcal E(ρ). We prove that for G compact, \mathcal O_\mathcal E(ρ) is strong Morita equivalent to a graph C^*-algebra. If λ is the left regular representation of an infinite, discrete and amenable group G, we show that \mathcal O_\mathcal E(λ) 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(ρ) with the crossed product C^*-correspondence.