Functions of positive type and GNS construction

Abstract

Two of the most important notions for unitary group representations are functions of positive type and induced representations, defined respectively in the present appendix and in Appendix E. We first discuss two kinds of kernels on a topological space X : those of positive type and those conditionally of negative type; the crucial difference is the presence or not of “conditionally”, whereas the difference between “positive” and “negative” is only a matter of sign convention. For each of these two types of kernels, there is a so-called GNS construction (for Gelfand, Naimark, and Segal) which shows how kernels are simply related to appropriate mappings of X to Hilbert spaces. Moreover, a theorem of Schoenberg establishes a relation between the two types of kernels. A function φ on a topological group G is of positive type if the kernel ( g , h ) ↦ φ( g −1 h ) is of positive type. Functions of positive type provide an efficient tool to prove some basic general results, such as the Gelfand–Raikov Theorem according to which a locally compact group has sufficiently many irreducible unitary representations to separate its points.