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. ${G^ \times }$ is the space of continuous finite-dimensional unitary representations of G. Let ${\operatorname {Hom}}({G^ \times },{H^ \times })$ denote the space of all continuous maps from ${G^ \times }$ to ${H^ \times }$ which preserve degree, direct sum, tensor product and equivalence. We prove that if H satisfies unitary duality, then ${\operatorname {Hom}}(G,H)$ and ${\operatorname {Hom}}{\mkern 1mu} ({H^ \times },{G^ \times })$ are naturally homeomorphic. Conversely, if ${\operatorname {Hom}}(Z,H)$ and ${\operatorname {Hom}}{\mkern 1mu} ({H^ \times },{Z^ \times })$ 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 ${\operatorname {Hom}}{\mkern 1mu} ({G^ \times },{H^ \times })$. 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.