Structure de certaines C * -algèbres associées aux réseaux de PSL 2 ()  [ On the structure of certain C * -algebra associated to lattices of PSL 2 () ]
Annales de l'Institut Fourier, Volume 52 (2002) no. 5, p. 1287-1299

By using the infinitesimal structure of the unitary irreducible representations of PSL 2 (), we give a complete description of certain C * -algebras associated to lattices in PSL 2 (); this gives answers to some questions of Bekka–de La Harpe–Valette.

En utilisant la structure infinitésimale des représentations unitaires irréductibles de PSL 2 (), nous donnons une description complète de certaines C * - algèbres associées aux réseaux de PSL 2 (), répondant ainsi à certaines questions de Bekka–de La Harpe–Valette.

DOI : https://doi.org/10.5802/aif.1919
Classification:  22D25,  43A15
Keywords: C * -algebras, unitary representations, (g,k)-modules, lattices
@article{AIF_2002__52_5_1287_0,
     author = {Pierrot, Fran\c cois},
     title = {Structure de certaines $C^*$-alg\`ebres associ\'ees aux r\'eseaux de ${\rm PSL}\_2({\mathbb {R}})$},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {52},
     number = {5},
     year = {2002},
     pages = {1287-1299},
     doi = {10.5802/aif.1919},
     zbl = {1053.22004},
     mrnumber = {1935551},
     language = {fr},
     url = {http://www.numdam.org/item/AIF_2002__52_5_1287_0}
}
Pierrot, François. Structure de certaines $C^*$-algèbres associées aux réseaux de ${\rm PSL}_2({\mathbb {R}})$. Annales de l'Institut Fourier, Volume 52 (2002) no. 5, pp. 1287-1299. doi : 10.5802/aif.1919. http://www.numdam.org/item/AIF_2002__52_5_1287_0/

[A] H. Araki Expansionals in Banach Algebra, Ann. Sci. de l'École Normale Supérieure, 4e série, Tome 6 (1973), pp. 67-84 | Numdam | MR 435842 | Zbl 0257.46054

[B] M. Bekka Restrictions of unitary representations to lattices and associated C * -algebras, JFA, Tome 143 (1997) | MR 1428115 | Zbl 0883.22006

[Ba] S. Baaj Multiplicateurs non bornés (1980) (Thèse de 3ème cycle, Paris VI)

[BCH] M. Bekka; M. Cowling; P. De La Harpe Some groups whose reduced C * -algebra is simple, Publ. Math. IHES, Tome 80 (1994), pp. 117-134 | Numdam | MR 1320606 | Zbl 0827.22001

[BH] M. Bekka; P. De La Harpe Représentations d'un groupe faiblement équivalentes à la représentation régulière, Bull. Soc. Math. France, Tome 122 (1994), pp. 333-342 | Numdam | MR 1294459 | Zbl 0824.22011

[BV] M. Bekka; A. Valette Lattices in semi-simple Lie groups and multipliers of group C * -algebras, SMF (Astérisque) Tome 232 (1995), pp. 67-79 | Zbl 0851.22006

[CS] M. Cowling; T. Steger The irreducibility of restrictions of unitary representations to lattices, J. Reine. Angew. Math, Tome 420 (1991), pp. 85-98 | MR 1124567 | Zbl 0760.22014

[D] J. Dixmier Les C * -algèbres et leurs représentations, Gauthiers-Villars (1964) | MR 171173 | Zbl 0152.32902

[K] G. Kasparov Lorentz groups: K-theory of unitary representations and crossed products, Soviet. Math. Dokl, Tome 29 (1984) | MR 741223 | Zbl 0584.22004

[L] S. Lang SL 2 (), Addison-Wesley (1975) | MR 430163

[Lan] E.C. Lance Hilbert C * -modules, a toolkit for operator algebraists, London Math. Soc. Lect. Notes Series, Tome 210 | MR 1325694 | Zbl 0822.46080

[M] D. Milicic Topological representations of the group C * -algebra of SL 2 (), Glas. Mat, Tome 6 (26) (1971), pp. 231-246 | MR 308795 | Zbl 0229.22010

[Ma] G.A. Margulis Discrete subgroups of semisimple Lie groups, Springer Verlag (1989) | MR 1090825 | Zbl 0732.22008

[P] F. Pierrot (2000) (Thèse de doctorat, Paris VII)

[Ri] M.A. Rieffel Induced representations of C * -algebras, Adv. in Math, Tome 13 (1974), pp. 176-257 | Article | MR 353003 | Zbl 0284.46040

[V] A. Valette Notes on the structure and the K-theory of the C * -algebra associated to SL 2 (}}R), Bull. Soc. Math. Belg. Sér. B, Tome XXXVI (1984), pp. 29-56 | MR 885552 | Zbl 0545.22003

[Vo] D. Voiculescu A non-commutative Weyl-Von Neumann theorem, Rev. R. Maths. Pures Appl, Tome 21 (1976), pp. 97-113 | MR 415338 | Zbl 0335.46039

[W] N. Wallach On the Selberg trace formula in the case of compact quotient, Bull. AMS, Tome vol 62 (1976) no. 2, pp. 171-195 | Article | MR 404533 | Zbl 0351.22008

[W2] N. Wallach Real reductive groups I, Academic Press (1988) | MR 929683 | Zbl 0666.22002

[Wo] S. L. Woronowicz Unbounded elements affiliated with C * -algebras and noncompact quantum groups, Comm. Math. Phys, Tome 136 (1991), pp. 399-432 | Article | MR 1096123 | Zbl 0743.46080