Locally compact quantum groups in the von Neumann algebraic setting

TL;DR

This paper advances the theory of locally compact quantum groups by providing a von Neumann algebraic definition, establishing connections to C*-algebraic quantum groups, and strengthening foundational properties like Haar weight invariance.

Contribution

It introduces a von Neumann algebraic framework for locally compact quantum groups and derives new results linking the quantum group with its dual, enhancing theoretical understanding.

Findings

Defined locally compact quantum groups in the von Neumann algebraic setting

Strengthened the invariance properties of the Haar weight

Connected the antipode with the dual's modular structures

Abstract

In this paper we complete in several aspects the picture of locally compact quantum groups. First of all we give a definition of a locally compact quantum group in the von Neumann algebraic setting and show how to deduce from it a C*-algebraic quantum group. Further we prove several results about locally compact quantum groups which are important for applications, but were not yet settled in our paper "Locally compact quantum groups". We prove a serious strengthening of the left invariance of the Haar weight, and we give several formulas connecting the locally compact quantum group with its dual. Loosely speaking we show how the antipode of the locally compact quantum group determines the modular group and modular conjugation of the dual locally compact quantum group.