Nonunimodular ring groups and Hopf--von~Neumann algebras

L. I. Vainerman ; G. I. Kats

Sbornik. Mathematics, Tome 23 (1974) no. 2, pp. 185-214

Résumé

A number of authors have introduced ring groups as objects generalizing locally compact groups. An analogue of the Pontryagin principle of duality holds for ring groups. In this paper we introduce a wider class of ring groups, one including the locally compact groups. A construction is given whereby to each ring group $\mathfrak G$ there is defined a dual ring group $\widehat{\mathfrak G}$; here $\widehat{\widehat{\mathfrak G}}=\mathfrak G$. By definition a ring group is determined by a $W^*$-algebra $\mathfrak A$ (the space of the ring group) equipped with an additional structure which allows $ \mathfrak A$ to be considered, in particular, as a Hopf–von Neumann algebra. When $\mathfrak G$ is a locally compact group, $\mathfrak A$ is the $W^*$-algebra of bounded measurable functions on $\mathfrak G$, considered in the natural way as operators in $L_2(\mathfrak G)$. Bibliography: 15 titles.