Bornological quantum groups as locally compact quantum groups

TL;DR

This paper demonstrates that bornological quantum groups can be used to construct and analyze locally compact quantum groups, extending the theory and providing new tools for subgroup analysis.

Contribution

It proves that bornological quantum groups give rise to locally compact quantum groups and shares many properties with algebraic quantum groups, offering new analytical tools.

Findings

Bornological quantum groups induce locally compact quantum groups.

They retain many properties of algebraic quantum groups.

Bornological subgroups correspond to closed subgroups in locally compact quantum groups.

Abstract

Bornological quantum groups were introduced by Voigt in order to generalize the theory of algebraic quantum groups in the sense of van Daele. In particular the class of bornological quantum groups contains all classical locally compact groups. In this paper we prove that a bornological quantum group gives rise to a locally compact quantum group, in a similar way to Kustermans and van Daele's result for algebraic quantum groups. We show that the bornological quantum groups, although more general than the algebraic ones, share most of their nice properties. We also argue that bornological quantum groups, when they occur as dense subalgebras of locally compact quantum groups, are useful tools for studying locally compact quantum groups. For instance, we show that the simple definition of a bornological closed quantum subgroup yields a closed subgroup of the locally compact quantum group in the sense of Vaes or Wooronowicz.