## Rocky Mountain Journal of Mathematics

### Cuntz-Pimsner algebras of group representations

Valentin Deaconu

#### 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.

#### Article information

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

Dates
First available in Project Euclid: 24 November 2018

https://projecteuclid.org/euclid.rmjm/1543028440

Digital Object Identifier
doi:10.1216/RMJ-2018-48-6-1829

Mathematical Reviews number (MathSciNet)
MR3879304

Zentralblatt MATH identifier
06987227

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

#### Citation

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. https://projecteuclid.org/euclid.rmjm/1543028440

#### References

• D.V. Benson, A. Kumjian and N.C. Phillips, Symmetries of Kirchberg algebras, Canad. Math. Bull. 46 (2003), 509–528.
• V. Deaconu, Group actions on graphs and $C^*$-correspondences, Houston J. Math. 44 (2018), 147–168.
• V. Deaconu, A. Kumjian, D. Pask and A. Sims, Graphs of $C^*$-correspondences and Fell bundles, Indiana Univ. Math. J. 59 (2010), 1687–1735.
• G. Hao and C.-K. Ng, Crossed products of $C^*$-correspondences by amenable group actions, J. Math. Anal. Appl. 345 (2008), 702–707.
• S. Kaliszewski, N.S. Larsen and J. Quigg, Subgroup correspondences, arXiv: 1612.04243v2.
• S. Kaliszewski, N. Patani and J. Quigg, Characterizing graph $C^*$-correspondences, Houston J. Math. 38 (2012), 751–759.
• T. Katsura, The ideal structures of crossed products of Cuntz algebras by quasi-free actions of abelian groups, Canad. J. Math. 55 (2003), 1302–1338.
• ––––, A construction of actions on Kirchberg algebras which induce given actions on their $K$-groups, J. reine angew. Math. 617 (2008), 27–65.
• A. Kishimoto and A. Kumjian, Crossed products of Cuntz algebras by quasi-free automorphisms, Fields Inst. Comm. 13 (1997), 173–192.
• A. Kumjian, On certain Cuntz-Pimsner algebras, Pacific J. Math. 217 (2004), 275–289.
• J. McKay, Graphs, singularities, and finite groups, Proc. Symp. Pure Math. 37 (1980), 183–186.
• M.H. Mann, I. Raeburn and C.E. Sutherland, Representations of finite groups and Cuntz-Krieger algebras, Bull. Australian Math. Soc. 46 (1992), 225–243.
• P. Muhly and B. Solel, On the Morita equivalence of tensor algebras, Proc. London Math. Soc. 81 (2000), 113–168.
• I. Raeburn and W. Szyma\' nski, Cuntz-Krieger algebras of infinite graphs and matrices, Trans. Amer. Math. Soc. 356 (2004), 39–59.
• J. Spielberg, Non-cyclotomic presentations of modules and prime-order automorphisms of Kirchberg algebras, J. reine angew. Math. 613 (2007), 211–230.
• M. Sugiura, Unitary representations and harmonic analysis, An introduction, second edition, North-Holland, Amsterdam, 1990.
• D. Williams, Crossed products of $C^*$-algebras, American Mathematical Society, Providence, RI, 2007.