A characterization of unitary duality

Abstract

The concept of unitary duality for topological groups was introduced by H. Chu. All mapping spaces are given the compact-open topology. Let G and H be locally compact groups. Gx is the space of continuous finite-dimensional unitary of G. Let Hom (Gx, Hx) denote the space of all continuous maps from Gx to Hx which preserve degree, direct sum, tensor product and equivalence. We prove that if H satisfies unitary duality, then Hom (G, H) and Hom (Hx, G x) are naturally homeomorphic. Conversely, if Hom (Z, H) and Hom (Hx, Zx) are homeomorphic by the natural map, where Z denotes the integers, then H satisfies unitary duality. In different contexts, results similar to the first half of this theorem have been obtained by Suzuki and by Ernest. The proof relies heavily on another result in this paper which gives an explicit characterization of the topology on Hom (G x, Hx ). In addition, we give another necessary condition for locally compact groups to satisfy unitary duality and use this condition to present an example of a maximally almost periodic discrete group which does not satisfy unitary duality. Introduction. Let G be a locally compact topological group. Let GX denote the space of continuous finite-dimensional unitary of G with the compact-open topology. G x x is the space of continuous unitary representations of G x (see Definition 2 below), again with the compact-open topology. G x x becomes a topological group in a natural way and there is a canonical homomorphism from G to GX x. If this canonical map is a topological isomorphism we say that G satisfies unitary duality. This definition was introduced by Chu [2]. It presents in a more general context the duality theorems of Pontryagin [9, p. 94 ff.], Tannaka [7], and Takahashi [6]. In ?1 of this paper we give the important definitions related to unitary duality. Hom (G x, H X) for two locally compact groups G and H is defined in a natural way and given the compact-open topology also. In ?2 we give an explicit subbase for the neighborhoods of each point in Hom (G X, H x). In ?3 we use the result of ?2 to prove the following characterization of unitary duality: if G and H are locally Presented in part to the Society, January 22, 1970 under the title Unitary duality of locally compact groups. II; received by the editors July 9, 1969. AMS Subject Classifications. Primary 2220; Secondary 2260.