Vol. 89, No. 2, 1980

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An analogue of the Wiener-Tauberian theorem for spherical transforms on semisimple Lie groups

Alladi Sitaram

Vol. 89 (1980), No. 2, 439–445

Let G be a semi-simple connected noncompact Lie group with finite center and K a fixed maximal compact subgroup of G. Fix a Haar measure dx on G and let I1(G) denote those functions in L1(G,dx) which are biinvariant under K. The purpose of this paper is to prove that if f I1(G) is such that its spherical Fourier transform (i.e., Gelfand transform) f is nowhere vanishing on the maximal ideal space of I1(G) and f “does not vanish too fast at ”, then the ideal generated by f is dense in I1(G). This generalizes earlier results of Ehrenpreis-Mautner for G = SL(2,R) and R. Krier for G of real rank one.

Mathematical Subject Classification 2000
Primary: 43A80
Secondary: 43A20
Received: 9 January 1979
Published: 1 August 1980
Alladi Sitaram