On Quantum Duality of Group Amenability

Xia Zhang,Ming Liu

doi:10.3390/sym12010085

Abstract

In this paper, we investigate the co-amenability of compact quantum groups. Combining with some properties of regular C*-norms on algebraic compact quantum groups, we show that the quantum double of co-amenable compact quantum groups is unique. Based on this, this paper proves that co-amenability is preserved under formulation of the quantum double construction of compact quantum groups, which exhibits a type of nice symmetry between the co-amenability of quantum groups and the amenability of groups.

Highlights

IntroductionGiven a compact group G, denoted by C ( G ) the C*-algebra of continuous functions on G, one can define a morphism
The morphism ∆ is called a co-multiplication on C ( G ), under which the pair (C ( G ), ∆) comes into being a compact quantum group defined in the sense of Woronowicz [1]
Under what conditions is the Haar integral on a compact quantum group (CQG) faithful and the co-unit norm-bounded? In [3], Bédos, Murphy, and Tuset defined the co-amenability of CQG, which can induce the faithfulness of its Haar integral and the norm-boundness of its co-unit

Summary

Introduction

Given a compact group G, denoted by C ( G ) the C*-algebra of continuous functions on G, one can define a morphism. The morphism ∆ is called a co-multiplication on C ( G ), under which the pair (C ( G ), ∆) comes into being a compact quantum group defined in the sense of Woronowicz [1]. The pair ( A, ∆) is called a compact quantum group (CQG). From [3], the Haar integral of (C ∗ (Γ), ∆) is faithful if and only if the co-unit of (C ∗ (Γ), ∆) is norm-bounded if and only if Γ is amenable. Under what conditions is the Haar integral on a CQG faithful and the co-unit norm-bounded? In [3], Bédos, Murphy, and Tuset defined the co-amenability of CQG, which can induce the faithfulness of its Haar integral and the norm-boundness of its co-unit.

Preliminaries

The Main Results

Conclusions