Negative definite functions (all definitions are given in § 1 below) on a locally compact, σ-compact group G have been used in several different contexts recently [2, 5, 7, 11]. In this paper we show how such functions relate to other properties such a group may have. Here are six properties which G might have. They are grouped into three pairs with one property of each pair involving negative definite functions. We show that the paired properties are equivalent and, where possible, give counter-examples to other equivalences. We assume throughout that G is not compact.(1A) G does not have property T.(IB) There is a continuous, negative definite function on G which is unbounded.(2A) G has the (weak and/or strong) dual R-L property.(2B) For every closed, non-compact set Q ⊂ G there is a continuous, negative definite function on G which is unbounded on Q.

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Abstract. For locally compact groups, Fourier algebras and Fourier-Stieltjes algebras have proved to be useful dual objects. They encode the representation theory of the group via the positive de nite functions on the group: positive de nite functions correspond to cyclic representations and span these algebras as linear spaces. They encode information about the algebra of the group in the geometry of the Banach space structure, and the group appears as a topological subspace of the maximal ideal space of the algebra W1, W2]. Because groupoids and their representations appear in studying operator algebras, ergodic theory, geometry, and the representation theory of groups, it would be useful to have a duality theory for them. This paper gives a rst step toward extending the theory of Fourier-Stieltjes algebras from groups to groupoids. If G is a locally compact (second countable) groupoid, we show that

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