Multiplicative partial isometries, manageability, and C*-algebraic quantum groupoids

TL;DR

This paper introduces the concept of multiplicative partial isometries, extending multiplicative unitaries, and shows how they can generate C*-algebraic quantum groupoids under certain conditions like manageability.

Contribution

It generalizes multiplicative unitaries to partial isometries and establishes conditions for constructing quantum groupoids from them.

Findings

Defined multiplicative partial isometries including the pentagon equation.

Under manageability, constructed C*-algebraic quantum groupoids.

Extended the framework of quantum groups to more general structures.

Abstract

Generalizing the notion of a multiplicative unitary (in the sense of Baaj-Skandalis), which plays a fundamental role in the theory of locally compact quantum groups, we develop in this paper the notion of a multiplicative partial isometry. The axioms include the pentagon equation, but more is needed. Under suitable conditions (such as the "manageability"), it is possible to construct from it a pair of C*-algebras having the structure of a C*-algebraic quantum groupoid of separable type. Generalizing the notion of a multiplicative unitary operator, which plays a fundamental role in the theory of locally compact quantum groups, we develop in this paper the notion of a multiplicative partial isometry. The axioms include the pentagon equation, but more is needed. Under the "manageability" condition on a multiplicative partial isometry (modified from the Woronowicz's condition for a multiplicative unitary), it is possible to construct from it a pair of C*-algebras having almost the structure of a C*-algebraic quantum groupoid of separable type.