On an analogue of the Wiener Tauberian theorem for symmetric spaces of the noncompact type

Abstract

Let X be a symmetric space and G the connected component of the group of isometries of X. If / G Lι (X), we consider conditions under which Sp{*/: g e G} is dense in Lι(X) in terms of the of /. This continues earlier work on this kind of problem by L. Ehrenpreis and F. I. Mautner, R. Krier and the author. 1. Introduction. Let /be an integrable function on R. Then we have the famous Wiener-Tauberian theorem: If the Fourier transform / is a nowhere vanishing function on i?, then the ideal generated by / is dense in Lι(R). In [EMI] Ehrenpreis and Mautner observed that the exact analogue of the above theorem is no longer true if one considers the commutative Banach algebra of ^-bi-invarian t Lι-functions on a non-compact semi-simple Lie group G, where K is a maximal compact subgroup of G—i.e. Let I (G) denote the commutative Banach algebra of Λ^-bi-invaria nt Lι-functions on G. For / e I\(G) let / denote its spherical Fourier transform (see §2). Then there exist functions / G I\(G) such that / is nowhere vanishing on the maximal ideal space M of I (G) and yet the algebra generated by / is not dense in h{G). However when G = SL(2, i?) they were able to show that a modified version of Wiener's theorem is true i.e. / nowhere vanishing on M together with the condition that it does not go to zero too fast at oo would indeed imply that the ideal generated by / is dense in I\(G). (Theorems 6 and 7 of [EMI].) This kind of result has been generalized by R. Krier [Kl] when G is a non-compact connected semisimple Lie group of real rank 1 and by the author for G of arbitrary rank [Si].