Positive-definite functions on infinite-dimensional groups

Abstract

This paper concerns positive-definite functions $\phi$ on infinite-dimensional groups G. Our main results are as follows: first, we claim that if G has a σ-finite measure μ on the Borel field $\frak{B}(G)$ whose right admissible shifts form a dense subgroup G ₀, a unique (up to equivalence) unitary representation (H, T) with a cyclic vector corresponds to $\phi$ through a method similar to that used for the G–N–S construction. Second, we show that the result remains true, even if we go to the inductive limits of such groups, and we derive two kinds of theorems, those taking either G or G ₀ as a central object. Finally, we proceed to an important example of infinite-dimensional groups, the group of diffeomorphisms ${\rm Diff}_0^*(M)$ on smooth manifolds M, and see that the correspondence between positive-definite functions and unitary representations holds for ${\rm Diff}_0^*(M)$ under a fairy mild condition. For a technical reason, we impose condition (c) in Sect. 2 on the measure space $(G,\frak{B}(G),\mu)$ throughout this paper. It is also a weak condition, and it is satified, if G is separable, or if μ is Radon.