Duality of Hopf $C^*$-algebras

TL;DR

This paper explores the duality theory of Hopf $C^*$-algebras, introducing a Hilbert-space-free framework and defining a Fourier algebra to analyze full and reduced duality.

Contribution

It presents a novel Hilbert-space-free approach to duality in Hopf $C^*$-algebras and introduces the Fourier algebra concept for these structures.

Findings

Defined the Fourier algebra of a Hopf $C^*$-algebra.

Analyzed full and reduced duality in a new framework.

Provided insights into duality without relying on Hilbert space representations.

Abstract

In this paper, we study the duality theory of Hopf $C^*$-algebras in a general ``Hilbert-space-free'' framework. Our particular interests are the ``full duality'' and the ``reduced duality''. In order to study the reduced duality, we define the interesting notion of Fourier algebra of a general Hopf $C^*$-algebra.