The full group C*-algebra of the modular group is primitive

Bédos, Erik; Omland, Tron

doi:10.1090/S0002-9939-2011-11143-2

The full group C$^*$-algebra of the modular group is primitive
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by Erik Bédos and Tron Å. Omland PDF

Proc. Amer. Math. Soc. 140 (2012), 1403-1411 Request permission

Abstract:

We show that the full group C$^*$-algebra of $PSL(n, \mathbb {Z})$ is primitive when $n=2$ and not primitive when $n\geq 3$. Moreover, we show that there exists an uncountable family of pairwise inequivalent, faithful irreducible representations of $C^*(PSL(2,\mathbb {Z}))$.

References

M. Bekka, M. Cowling, and P. de la Harpe, Simplicity of the reduced $C^*$-algebra of $\textrm {PSL}(n,\textbf {Z})$, Internat. Math. Res. Notices 7 (1994), 285ff., approx. 7 pp.}, issn=1073-7928, review= MR 1283024, doi=10.1155/S1073792894000322,
M. Bekka, M. Cowling, and P. de la Harpe, Some groups whose reduced $C^*$-algebra is simple, Inst. Hautes Études Sci. Publ. Math. 80 (1994), 117–134 (1995). MR 1320606
Bachir Bekka and Pierre de la Harpe, Irreducibly represented groups, Comment. Math. Helv. 83 (2008), no. 4, 847–868. MR 2442965, DOI 10.4171/CMH/145
Bachir Bekka, Pierre de la Harpe, and Alain Valette, Kazhdan’s property (T), New Mathematical Monographs, vol. 11, Cambridge University Press, Cambridge, 2008. MR 2415834, DOI 10.1017/CBO9780511542749
Man Duen Choi, The full $C^{\ast }$-algebra of the free group on two generators, Pacific J. Math. 87 (1980), no. 1, 41–48. MR 590864
Kenneth R. Davidson, $C^*$-algebras by example, Fields Institute Monographs, vol. 6, American Mathematical Society, Providence, RI, 1996. MR 1402012, DOI 10.1090/fim/006
Jacques Dixmier, Les $C^{\ast }$-algèbres et leurs représentations, Cahiers Scientifiques, Fasc. XXIX, Gauthier-Villars Éditeur, Paris, 1969 (French). Deuxième édition. MR 0246136
Pierre de la Harpe, On simplicity of reduced $C^\ast$-algebras of groups, Bull. Lond. Math. Soc. 39 (2007), no. 1, 1–26. MR 2303514, DOI 10.1112/blms/bdl014
N. Khattou, Deux propriétés de la $C^*$-algèbre maximale de certains produits libres, Comptes rendus de la première recontre maroco-andalouse sur les algèbres et leurs applications (Tétouan, 2001) Univ. Abdelmalek Essaâdi. Fac. Sci. Tétouan, Tétouan, 2003, pp. 54–63 (French). MR 2029849
Eberhard Kirchberg, On nonsemisplit extensions, tensor products and exactness of group $C^*$-algebras, Invent. Math. 112 (1993), no. 3, 449–489. MR 1218321, DOI 10.1007/BF01232444
George W. Mackey, Borel structure in groups and their duals, Trans. Amer. Math. Soc. 85 (1957), 134–165. MR 89999, DOI 10.1090/S0002-9947-1957-0089999-2
George W. Mackey, Unitary representations of group extensions. I, Acta Math. 99 (1958), 265–311. MR 98328, DOI 10.1007/BF02392428
George W. Mackey, Induced representations and normal subgroups, Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960) Jerusalem Academic Press, Jerusalem; Pergamon, Oxford, 1961, pp. 319–326. MR 0139687
G. J. Murphy, Primitivity conditions for full group $C^\ast$-algebras, Bull. London Math. Soc. 35 (2003), no. 5, 697–705. MR 1989500, DOI 10.1112/S0024609303002352
Judith A. Packer, Twisted group $C^*$-algebras corresponding to nilpotent discrete groups, Math. Scand. 64 (1989), no. 1, 109–122. MR 1036431, DOI 10.7146/math.scand.a-12250
Judith A. Packer and Iain Raeburn, Twisted crossed products of $C^*$-algebras, Math. Proc. Cambridge Philos. Soc. 106 (1989), no. 2, 293–311. MR 1002543, DOI 10.1017/S0305004100078129
Gert K. Pedersen, $C^{\ast }$-algebras and their automorphism groups, London Mathematical Society Monographs, vol. 14, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1979. MR 548006
John C. Quigg, On the irreducibility of an induced representation, Pacific J. Math. 93 (1981), no. 1, 163–179. MR 621605
John C. Quigg, On the irreducibility of an induced representation. II, Proc. Amer. Math. Soc. 86 (1982), no. 2, 345–348. MR 667304, DOI 10.1090/S0002-9939-1982-0667304-8
Jean-Pierre Serre, Trees, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. Translated from the French original by John Stillwell; Corrected 2nd printing of the 1980 English translation. MR 1954121
Hisaaki Yoshizawa, Some remarks on unitary representations of the free group, Osaka Math. J. 3 (1951), 55–63. MR 41854
G. Zeller-Meier, Produits croisés d’une $C^{\ast }$-algèbre par un groupe d’automorphismes, J. Math. Pures Appl. (9) 47 (1968), 101–239 (French). MR 241994