A class of C*-algebraic locally compact quantum groupoids Part I. Motivation and definition

TL;DR

This paper introduces a new class of locally compact quantum groupoids within the C*-algebra framework, motivated by algebraic structures like weak multiplier Hopf algebras, and establishes their foundational properties.

Contribution

It formulates the definition of these quantum groupoids, highlighting the role of a canonical idempotent and extending the theory of locally compact quantum groups.

Findings

Definition of C*-algebraic locally compact quantum groupoids

Existence and role of a canonical idempotent element

Framework includes locally compact quantum groups as a special case

Abstract

In this series of papers, we develop the theory of a class of locally compact quantum groupoids, which is motivated by the purely algebraic notion of weak multiplier Hopf algebras. In this Part I, we provide motivation and formulate the definition in the C*-algebra framework. Existence of a certain canonical idempotent element is required and it plays a fundamental role, including the establishment of the coassociativity of the comultiplication. This class contains locally compact quantum groups as a subclass.