Locally compact quantum groups in the universal setting

TL;DR

This paper constructs a universal C*-algebraic quantum group for each reduced quantum group, demonstrating that the universal version retains a rich quantum group structure and generalizes prior results.

Contribution

It introduces a universal C*-algebraic quantum group associated with every reduced quantum group, extending the framework of quantum group theory.

Findings

Universal C*-algebra carries a quantum group structure

Every *-representation of a modified L1-space is generated by a unitary corepresentation

Universal quantum group is as rich as the reduced one

Abstract

In this paper we associate to every reduced C*-algebraic quantum group A a universal C*-algebraic quantum group. We fine tune a proof of Kirchberg to show that every *-representation of a modified L1-space is generated by a unitary corepresentation. By taking the universal enveloping C*-algebra of a dense sub *-algebra of A we arrive at the uinversal C*-algebra. We show that this universal C*-algebra carries a quantum group structure which is as rich as its reduced companion.