The spherical transform on projective limits of symmetric spaces

The theory of a spherical Fourier transform for measures on certain projective limits of symmetric spaces of non-compact type is developed. Such spaces are introduced for the first time and basic properties of the spherical transform, including a Levy-Cramer type continuity theorem, are obtained. The results are applied to obtain a heat kernel measure on the limit space which is shown to satisfy a certain cylindrical heat equation. The projective systems under consideration arise from direct systems of semi-simple Lie groups {lcub}Gj{rcub} such that Gj is essentially the semi-simple component of a parabolic subgroup of Gj+1. This class includes most of the classical families of Lie groups as well as infinite direct products of semi-simple groups. In the case each Gj is a complex group, a characterization of the image of the spherical transform is given.