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 \(\text{Bin}(\Gamma)\).


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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