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Property \(T\) for discrete groups in terms of their regular representation. (English) Zbl 0787.22007

The main result of the article is a generalization of a theorem of A. Connes and V. Jones on Kazhdan’s property \(T\) for discrete groups: it is proved (without any assumption on conjugacy classes) that a group \(\Gamma\) has property \(T\) if and only if the identity correspondence \(L^ 2(M)\) of the von Neumann algebra \(M\) of \(\Gamma\) admits a neighbourhood \(U\) such that any correspondence belonging to \(U\) contains some non zero subcorrespondence of \(L^ 2(M)\). Moreover, a new ideal \(\text{Bin}(\Gamma)\) of the Fourier-Stieltjes algebra \(B(\Gamma\times\Gamma)\) is introduced: It is the set of elements \(\varphi\) of \(B(\Gamma\times\Gamma)\) such that the one variable functions \(\varphi(\cdot,\gamma)\) and \(\varphi(\beta,\cdot)\) belong to the Fourier algebra \(A(\Gamma)\) for all fixed \(\beta\) and \(\gamma\). It is proved that every element of \(\text{Bin}(\Gamma)\) comes from a coefficient of some correspondence of \(M\), and property \(T\) of \(\Gamma\) is also expressed in terms of positive definite functions belonging to \(\text{Bin}(\Gamma)\).

MSC:

22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
46L10 General theory of von Neumann algebras
43A35 Positive definite functions on groups, semigroups, etc.
43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis)
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References:

[1] Arsac, G.: Sur l’espace de Banach engendr? par les coefficients d’une repr?sentation unitaire. Publ. Math. Lyon13-2, 1-101 (1976) · Zbl 0365.43005
[2] Bekka, M.E.B., Valette, A.: Kazhdan’s property (T) and amenable representations. Math. Z.212, 293-299 (1993) · Zbl 0789.22006
[3] Connes, A., Jones, V.: PropertyT for von Neumann algebras. Bull. Lond. Math. Soc.17, 57-62 (1985) · Zbl 1190.46047
[4] Effros, E., Lance, E.: Tensor products of operator algebras. Adv. Math.25, 1-34 (1977) · Zbl 0372.46064
[5] Eymard, P.: L’alg?bre de Fourier d’un groupe localement compact. Bull. Soc. Math. Fr.,92, 181-236 (1964) · Zbl 0169.46403
[6] Gaal, S.: Linear analysis and representation theory. Berlin Heidelberg New York: Springer 1973 · Zbl 0275.43008
[7] Godement, R.: Les fonctions de type positif et la th?orie des groupes. Trans. Am. Math. Soc.63, 1-84 (1948) · Zbl 0031.35903
[8] Harpe, P. de la, Valette, A.: La propri?t? (T) de Kazhdan pour les groupes localement compacts. Ast?risque175 (1989) · Zbl 0759.22001
[9] Kallman, R.: A generalization of free action. Duke Math. J.36, 781-789 (1969) · Zbl 0184.17101
[10] Mingo, J.: The correspondence associated to an inner completely positive map. Math. Ann.284, 121-135 (1989) · Zbl 0646.46056
[11] Popa, S.: Correspondences. Preprint (1986)
[12] Robertson, A.G.: PropertyT for II1 factors and unitary representations of Kazhdan groups. Preprint (1992)
[13] Takesaki, M.: Theory of operator algebras I. Berlin Heidelberg New York, Springer 1979 · Zbl 0436.46043
[14] Walter, M.E.:W *-algebras and nonabelian harmonic analysis. J. Funct. Anal.11, 17-38 (1972) · Zbl 0242.22010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.