Duality and Subgroups

Abstract

The second author proved, in [27], a duality theorem for general locally compact groups as a generalization of the so-called Tannaka duality theorem for compact groups. In the proof, the regular representation plays an essential role. In order to clarify the role of the regular representation in duality theory, the first author gave, in [23], a characterization of group algebras in terms of a Hopf-von Neumann algebra and showed that the abelian von Neumann algebra L-(G) of all essentially bounded functions on a locally compact group G together with the co-multiplication 6, the involution j and the Haar measure f involves the duality principle for the given group G. In this paper we shall examine, by showing a correspondence between closed subgroups and subalgebras of the Hopf-von Neumann algebra {L-(G), a, j}, how this duality principle works for subgroups. For a locally compact abelian group G, the Pontryagin duality theorem shows that there is a beautiful and complete one-to-one correspondence between a closed subgroup H of G and a closed subgroup H I of the dual group G such that