Banach Journal of Mathematical Analysis

Primitivity of some full group C*-algebras

Erik Bedos and Tron A. Omland

Full-text: Open access


We show that the full group $C^*$-algebra of the free product of two nontrivial countable amenable discrete groups, where at least one of them has more than two elements, is primitive. We also show that in many cases, this $C^*$-algebra is antiliminary and has an uncountable family of pairwise inequivalent, faithful irreducible representations.

Article information

Banach J. Math. Anal., Volume 5, Number 2 (2011), 44-58.

First available in Project Euclid: 14 August 2011

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46L05: General theory of $C^*$-algebras
Secondary: 22D25: $C^*$-algebras and $W^*$-algebras in relation to group representations [See also 46Lxx] 46L55: Noncommutative dynamical systems [See also 28Dxx, 37Kxx, 37Lxx, 54H20]

full group C*-algebra primitivity free product antiliminary


Bedos, Erik; Omland, Tron A. Primitivity of some full group C*-algebras. Banach J. Math. Anal. 5 (2011), no. 2, 44--58. doi:10.15352/bjma/1313363001.

Export citation


  • E. Bédos and T. Omland, The full group $C^*$-algebra of the modular group is primitive, Proc. Amer. Math. Soc. (to appear).
  • P.-A. Cherix, M. Cowling, P. Jolissaint, P. Julg and A. Valette, Groups with the Haagerup Property, Gromov's a-T-menability, Progress in Mathematics, 197, Birkhäuser Verlag, Basel, 2001.
  • M.D. Choi, The full group $C^*$-algebra of the free group on two generators, Pacific J. Math. 87 (1980), 41–48.
  • K.J. Dykema and P. de la Harpe, Some groups whose reduced $C^*$-algebras have stable rank one, J. Math. Pures Appl. 78 (1999), 591–608.
  • R. Exel and T. Loring, Finite-dimensional representations of free product $C^*$-algebras, Internat. J. Math. 3 (1992), 469–476.
  • N. Higson and G. Kasparov, Operator K-theory for groups which act properly and isometrically on Hilbert space, Electron. Res. Announc. Amer. Math. Soc. 3 (1997), 131–142.
  • N. Khattou, Deux propriétés de la $C^*$-algèbre maximale de certains produits libres, Comptes rendus de la première rencontre maroco- andalouse sur les algèbres et leurs applications (Tétouan, 2001), 54–63, Univ. Abdelmalek Essaadi, Fac. Sci. Tétouan, Tétouan, 2003.
  • R. Li and S. Pedersen, Certain full group $C^*$-algebras without proper projections, J. Operator Theory 24 (1990), 239–253.
  • G.J. Murphy, Primitivity conditions for full group $C^*$-algebras, Bull. Lond. Math. Soc. 35 (2003), 697–705.
  • J.A. Packer, Twisted group $C^*$-algebras corresponding to nilpotent discrete groups, Math. Scand. 64 (1989), 109–122.
  • J.A. Packer and I. Raeburn, Twisted crossed products of $C^*$-algebras, Math. Proc. Camb. Phil. Soc. 106 (1989), 293–311.
  • G.K. Pedersen, $C^*$-Algebras and Their Automorphisms Groups. Academic Press, London, 1979.
  • J.P. Serre, Trees, Springer-Verlag, Berlin, 2003.
  • J.L. Tu, La conjecture de Baum–Connes pour les feuilletages moyennables, J. K-Theory 17 (1999), 215–264.
  • A. Valette, The conjecture of idempotents: a survey of the $C^*$-algebraic approach, Bull. Math. Soc. Belg. 41 (1989), 485–521.
  • H. Yoshizawa, Some remarks on unitary representations of the free group, Osaka Math. J. 3 (1951), 55–63.