Positive definite functions on infinite-dimensional convex cones

Abstract

Part I. Preliminaries and Preparatory Results: Bounded and unbounded operators Cone-valued measures Measures on topological spaces Projective limits of cone-valued measures Holomorphic functions Involutive semigroups and their representations Positive definite kernels and functions $\boldmath{C^*}$-algebras associated with involutive semigroups Integral representations of positive definite functions Convex cones and their faces Examples of convex cones Conelike semigroups: definition and examples Representations of conelike semigroups I Fourier and Laplace transforms Generalized Bochner and Stone Theorems Part II. Main Results: Nussbaum Theorem for open convex cones Positive definite functions on convex cones with non-empty interior Positive definite functions on convex sets Associated Hilbert spaces and representations Nussbaum Theorem for generating convex cones Representations of conelike semigroups II Associated unitary representations Holomorphic extension of unitary representations Holomorphic extension of representations of nuclear groups References Index List of symbols.