In this paper, we establish Bochner Weil type theorems and integral formulas of Le vy Khinchin type in the setting of locally compact commutative semigroups with involution. These results are used to prove some new holomorphic extension results. For a conelike involution semigroup S in a finite dimensional real space V, we establish integral characterizations for continuous positive definite and definitizable functions on S. In particular, we obtain holomorphic and Cauchy Riemann extensions of these functions by means of an integral representation. We prove that if z 0 is a point in the complexified vector space W :=V+iV and if a function is defined on z 0 +S then either positive definiteness or definitizability of f extends holomorphy from a domain 0 containing z 0 to a tube domain in W containing the sum of 0 and the convex cone spanned by S. We also obtain the holomorphic extension as a generalized Fourier Laplace transform of some measure or a Le vy Khinchin type integral representation.
1998 Academic Press
0. Introduction
An outstanding problem in harmonic analysis on semigroups is to know whether the classical Bochner Weil theorem has a generalization from the group setting to the context of locally compact semigroups with involution. Namely, given a locally compact commutative semigroup S with involution V and identity e and a positive definite function . on S, is it true that the function . is continuous at the identity e if and only if it is the generalized Fourier Laplace transform of a non-negative Radon measure on the space of F=F(S) of all those semicharacters on S which are continuous at e? This problem was raised in [BCR] and [Be1]. Recently, two partial solutions to this problem have been obtained in [Re] and [Yo1]. In his work [Re], Ressel introduced an interesting class of semigroups called conelike and solved the above problem for bounded positive definite functions on these semigroups. In [Yo1], the author considered the most general setting and introduced an admissibility assumption on the semigroup S which implies that the article no. FU973230