Multipliers, Self-Induced and Dual Banach Algebras

TL;DR

This paper surveys multiplier theory in Banach algebras, focusing on algebraic methods for extending module actions, and characterizes when multiplier algebras are dual Banach algebras, with applications to quantum groups.

Contribution

It provides an algebraic approach to multiplier theory, criteria for duality of multiplier algebras, and applies these to quantum group convolution algebras.

Findings

Multiplier algebra of a locally compact quantum group is always a dual Banach algebra.

A simple criterion for when a multiplier algebra is a dual Banach algebra.

Extended Haagerup tensor product can often be replaced by a multiplier algebra.

Abstract

In the first part of the paper, we present a short survey of the theory of multipliers, or double centralisers, of Banach algebras and completely contractive Banach algebras. Our approach is very algebraic: this is a deliberate attempt to separate essentially algebraic arguments from topological arguments. We concentrate upon the problem of how to extend module actions, and homomorphisms, from algebras to multiplier algebras. We then consider the special cases when we have a bounded approximate identity, and when our algebra is self-induced. In the second part of the paper, we mainly concentrate upon dual Banach algebras. We provide a simple criterion for when a multiplier algebra is a dual Banach algebra. This is applied to show that the multiplier algebra of the convolution algebra of a locally compact quantum group is always a dual Banach algebra. We also study this problem within the framework of abstract Pontryagin duality, and show that we construct the same weak$^*$-topology. We explore the notion of a Hopf convolution algebra, and show that in many cases, the use of the extended Haagerup tensor product can be replaced by a multiplier algebra.