Options
Valette, Alain
Résultat de la recherche
Kazhdan's property (T) : New mathematical monographs
2008, Bekka, Bachir, Valette, Alain, De La Harpe, Pierre
Lattices in semi-simple Lie groups and multipliers of group C*-algebras
1995, Bekka, Bachir, Valette, Alain
Let Gamma be a lattice in a non-compact simple Lie group G. We prove that the canonical map from the full C*-algebra C*(Gamma) to the multiplier algebra M(C*(G)) is not injective in general (it is never injective if G has Kazhdan's property (T), and not injective for many lattices either in SO(n, 1) or SU(n, 1)). For a locally compact group G, Fell introduced a property (WF3), stating that for any closed subgroup H of G, the canonical map from C*(H) to M(C*(G)) is injective. We prove that, for an almost connected G, property (WF3) is equivalent to amenability.
Group cohomology, harmonic functions and the first L
1997, Bekka, Bachir, Valette, Alain
For an infinite, finitely generated group Gamma, we study the first cohomology group H-1(Gamma, lambda(Gamma)) with coefficients in the left regular representation lambda(Gamma) of Gamma on l(2)(Gamma). We first prove that H-1(Gamma, C Gamma) embeds into H-1(Gamma, lambda(Gamma)); as a consequence, if H-1(Gamma, lambda(Gamma)) = 0, then Gamma is not amenable with one end. For a Cayley graph X of Gamma, denote by HD(X) the space of harmonic functions on X with finite Dirichlet sum. We show that, if Gamma is not amenable, then there is a natural isomorphism between H-1(Gamma, lambda(Gamma)) and HD(X)/C (the latter space being isomorphic to the first L-2-cohomology space of Gamma). We draw the following consequences: (1) If Gamma has infinitely many ends, then HD(X) not equal C; (2) If Gamma has Kazhdan's property (T), then HD(X) = C; (3) The property H-1(Gamma, lambda(Gamma)) = 0 is a quasi-isometry invariant; (4) Either H-1(Gamma, lambda(Gamma)) = 0 or H-1(Gamma, lambda(Gamma)) is infinite-dimensional; (5) If Gamma = Gamma(1) x Gamma(2) with Gamma(1) non-amenable and Gamma(2) infinite, then H-1(Gamma, lambda(Gamma)) = 0.
Kazhdan Property (T) and Amenable Representations
1993, Bekka, Bachir, Valette, Alain
Proper affine isometric actions of amenable groups
1995, Bekka, Bachir, Cherix, Pierre-Alain, Valette, Alain
On Duals of Lie-Groups Made Discrete
1993, Bekka, Bachir, Valette, Alain