Memoirs of the American Mathematical Society 2003; 128 pp; softcover Volume: 166 ISBN10: 0821832565 ISBN13: 9780821832561 List Price: US$59 Individual Members: US$35.40 Institutional Members: US$47.20 Order Code: MEMO/166/789
 This memoir is devoted to the study of positive definite functions on convex subsets of finite or infinitedimensional vector spaces, and to the study of representations of convex cones by positive operators on Hilbert spaces. Given a convex subset \(\Omega\subseteq V\) of a real vector space \(V\), we show that a function \(\phi\!:\Omega\to\mathbb{R}\) is the Laplace transform of a positive measure \(\mu\) on the algebraic dual space \(V^*\) if and only if \(\phi\) is continuous along line segments and positive definite. If \(V\) is a topological vector space and \(\Omega\subseteq V\) an open convex cone, or a convex cone with nonempty interior, we describe sufficient conditions for the existence of a representing measure \(\mu\) for \(\phi\) on the topological dual space\(V'\). The results are used to explore continuity properties of positive definite functions on convex cones, and their holomorphic extendibility to positive definite functions on the associated tubes \(\Omega+iV\subseteq V_{\mathbb{C}}\). We also study the interplay between positive definite functions and representations of convex cones, and derive various characterizations of those representations of convex cones on Hilbert spaces which are Laplace transforms of spectral measures. Furthermore, for scalar or operatorvalued positive definite functions which are Laplace transforms, we realize the associated reproducing kernel Hilbert space as an \(L^2\)space \(L^2(V^*,\mu)\) of vectorvalued functions and link the natural translation operators on the reproducing kernel space to multiplication operators on \(L^2(V^*,\mu)\), which gives us refined information concerning the norms of these operators.This memoir is devoted to the study of positive definite functions on convex subsets of finite or infinitedimensional vector spaces, and to the study of representations of convex cones by positive operators on Hilbert spaces. Given a convex subset \(\Omega\subseteq V\) of a real vector space \(V\), we show that a function \(\phi\!:\Omega\to\mathbb{R}\) is the Laplace transform of a positive measure \(\mu\) on the algebraic dual space \(V^*\) if and only if \(\phi\) is continuous along line segments and positive definite. If \(V\) is a topological vector space and \(\Omega\subseteq V\) an open convex cone, or a convex cone with nonempty interior, we describe sufficient conditions for the existence of a representing measure \(\mu\) for \(\phi\) on the topological dual space \(V'\). The results are used to explore continuity properties of positive definite functions on convex cones, and their holomorphic extendibility to positive definite functions on the associated tubes \(\Omega+iV\subseteq V_\mathbb C\). We also study the interplay between positive definite functions and representations of convex cones, and derive various characterizations of those representations of convex cones on Hilbert spaces which are Laplace transforms of spectral measures. Furthermore, for scalar or operatorvalued positive definite functions which are Laplace transforms, we realize the associated reproducing kernel Hilbert space as an \(L^2\)space \(L^2(V^*,\mu)\) of vectorvalued functions and link the natural translation operators on the reproducing kernel space to multiplication operators on \(L^2(V^*,\mu)\), which gives us refined information concerning the norms of these operators. Readership Graduate students and research mathematicians interested in analysis. Table of Contents Part I. Preliminaries and Preparatory Results  Bounded and unbounded operators
 Conevalued measures
 Measures on topological spaces
 Projective limits of conevalued measures
 Holomorphic functions
 Involutive semigroups and their representations
 Positive definite kernels and functions
 \(C^*\)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 nonempty 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
