Vol. 69, No. 2, 1977

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Positive definite functions which vanish at infinity

Alessandro Figà-Talamanca

Vol. 69 (1977), No. 2, 355–363

Let G be a separable noncompact locally compact group. Let A(G) and B(G), respectively, be the Fourier algebra and the Fourier-Stieltjes algebra of G as defined by P. Eymard. We prove that if G is unimodular and satisfies an additional hypothesis, which implies noncompactness, there exists an element of B(G), indeed a positive definite function, which vanishes at infinity but is not in A(G). This function actually belongs to Bρ(G), that is, it defines a unitary representation of G which is weakly contained in the regular representation.

Mathematical Subject Classification 2000
Primary: 43A30
Secondary: 43A35
Received: 2 October 1973
Revised: 13 October 1976
Published: 1 April 1977
Alessandro Figà-Talamanca