A class of C*-algebraic locally compact quantum groupoids Part II: Main theory

TL;DR

This paper develops the core theoretical framework of a specific subclass of locally compact quantum groupoids, focusing on their algebraic structure, regular representations, and antipode map, based on weak multiplier Hopf algebras.

Contribution

It introduces the main structural theory of quantum groupoids of separable type, including regular representations and the antipode, advancing the algebraic understanding of these objects.

Findings

Construction of right/left regular representations

Development of the antipode map

Elucidation of the structure of quantum groupoids

Abstract

This is Part II in our multi-part series of papers developing the theory of a subclass of locally compact quantum groupoids ("quantum groupoids of separable type"), based on the purely algebraic notion of weak multiplier Hopf algebras. The definition was given in Part I. The existence of a certain canonical idempotent element $E$ plays a central role. In this Part II, we develop the main theory, discussing the structure of our quantum groupoids. We will construct from the defining axioms the right/left regular representations and the antipode map.