Structure de certaines $C^{*}$-algèbres associées aux réseaux de ${\mathrm{PSL}}_2\left(ℝ\right)$

Pierrot, François

Annales de l'Institut Fourier, Tome 52 (2002) no. 5, pp. 1287-1299.

En utilisant la structure infinitésimale des représentations unitaires irréductibles de ${\mathrm{PSL}}_2\left(ℝ\right)$, nous donnons une description complète de certaines $C^{*}$- algèbres associées aux réseaux de ${\mathrm{PSL}}_2\left(ℝ\right)$, répondant ainsi à certaines questions de Bekka–de La Harpe–Valette.

By using the infinitesimal structure of the unitary irreducible representations of ${\mathrm{PSL}}_2\left(ℝ\right)$, we give a complete description of certain $C^{*}$-algebras associated to lattices in ${\mathrm{PSL}}_2\left(ℝ\right)$; this gives answers to some questions of Bekka–de La Harpe–Valette.

Classification : 22D25, 43A15
Mot clés : $C^*$-algèbres, représentations unitaires, $(g,k)$-modules, réseaux
Keywords: $C^*$-algebras, unitary representations, $(g,k)$-modules, lattices

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

[B] M. Bekka Restrictions of unitary representations to lattices and associated $C^{*}$-algebras, JFA, Volume 143 (1997) | MR | Zbl

[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, Volume 80 (1994), pp. 117-134 | Numdam | MR | Zbl

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

[BV] M. Bekka; A. Valette Lattices in semi-simple Lie groups and multipliers of group $C^{*}$-algebras (Astérisque), Volume 232 (1995), pp. 67-79 | Numdam | Zbl

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

[D] J. Dixmier Les $C^{*}$-algèbres et leurs représentations, Gauthiers-Villars, 1964 | MR | Zbl

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

[L] S. Lang $S{L}_2\left(ℝ\right)$, Addison-Wesley, 1975 | MR

[Lan] E.C. Lance Hilbert $C^{*}$-modules, a toolkit for operator algebraists, London Math. Soc. Lect. Notes Series, Volume 210 | MR | Zbl

[M] D. Milicic Topological representations of the group $C^{*}$-algebra of $S{L}_2\left(ℝ\right)$, Glas. Mat, Volume 6 (26) (1971), pp. 231-246 | MR | Zbl

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

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

[Ri] M.A. Rieffel Induced representations of $C^{*}$-algebras, Adv. in Math, Volume 13 (1974), pp. 176-257 | DOI | MR | Zbl

[V] A. Valette Notes on the structure and the K-theory of the $C^{*}$-algebra associated to $S{L}_2\left(\right\}\left\}R\right)$, Bull. Soc. Math. Belg. Sér. B, Volume XXXVI (1984), pp. 29-56 | MR | Zbl

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

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

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

[Wo] S. L. Woronowicz Unbounded elements affiliated with $C^{*}$-algebras and noncompact quantum groups, Comm. Math. Phys, Volume 136 (1991), pp. 399-432 | DOI | MR | Zbl