Bornological quantum groups as 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.