Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Byung-Jay Kahng

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.

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

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.

Paper Structure

This paper contains 14 sections, 51 theorems, 171 equations.

Table of Contents

  1. Introduction
  2. Multiplicative partial isometries
  3. The manageability condition
  4. Fullness condition
  5. Manageability condition
  6. The coalgebra structures on $A$ and $\widehat{A}$
  7. Antipode
  8. Unitary antipode
  9. Scaling group
  10. Antipode
  11. The base algebras
  12. The $C^*$-subalgebras $B$ and $C$
  13. The maps $\kappa$ and $\kappa'$
  14. Final remarks

Key Result

Proposition 1.2

Let $W$ be a multiplicative partial isometry. Consider the subspaces $B$, $C$, $\widehat{B}$, $\widehat{C}$ in ${\mathcal{B}}({\mathcal{H}})$ as above. We have:

Theorems & Definitions (121)

  • Definition 1.1
  • Remark
  • Proposition 1.2
  • proof
  • Proposition 1.3
  • proof
  • Remark
  • Lemma 1.4
  • proof
  • Remark
  • ...and 111 more