Polyhomomorphisms of locally compact groups

Yu. A. Neretin

Sbornik. Mathematics, Tome 212 (2021) no. 2, pp. 185-210

Résumé

Let $G$ and $H$ be locally compact groups with fixed two-sided invariant Haar measures. A polyhomomorphism $G\rightarrowtail H$ is a closed subgroup $R\subset G\times H$ with fixed Haar measure, whose marginals on $G$ and $H$ are dominated by the Haar measures on $G$ and $H$. A polyhomomorphism can be regarded as a multi-valued map sending points to sets equipped with ‘uniform’ measures. For two polyhomomorphisms $G\rightarrowtail H$ and $H\rightarrowtail K$ there is a well-defined product $G\rightarrowtail K$. The set of polyhomomorphisms $G\rightarrowtail H$ is a metrizable compact space with respect to the Chabauty-Bourbaki topology and the product is separately continuous. A polyhomomorphism $G\rightarrowtail H$ determines a canonical operator $L^2(H)\to L^2(G)$, 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.

Détail

Keywords: Haar measure, partial isometries, Chabauty-Bourbaki topology.
Mots-clés : polymorphism, multiplicative relation