Positive-definite functions on infinite-dimensional groups

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 sigma-finite measure mu on the Borel field 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 Diff(0)*(M) on smooth manifolds M, and see that the correspondence between positive-definite functions and unitary representations holds for Diff(0)*(M) under a fairy mild condition. For a technical reason, we impose condition (c) in Sect. 2 on the measure space (G, B(G), mu) throughout this paper. It is also a weak condition, and it is satified, if G is separable, or if mu is Radon.