Polyhomomorphisms of locally compact groups

Abstract

Abstract Let and be locally compact groups with fixed two-sided invariant Haar measures. A polyhomomorphism is a closed subgroup with fixed Haar measure, whose marginals on and are dominated by the Haar measures on and . A polyhomomorphism can be regarded as a multi-valued map sending points to sets equipped with ‘uniform’ measures. For two polyhomomorphisms and there is a well-defined product . The set of polyhomomorphisms is a metrizable compact space with respect to the Chabauty-Bourbaki topology and the product is separately continuous. A polyhomomorphism determines a canonical operator , which is a partial isometry up to a scalar factor. For example, we consider locally compact linear spaces over finite fields and examine the closures of groups of linear operators in semigroups of polyhomomorphisms. Bibliography: 40 titles.