Quasi-multipliers on weak Arens regular Banach algebras

M Adib

doi:10.12988/pms.2013.13006

Abstract

In this paper we define the notion of weak Arens regular Banach algebras and extend the concept of quasi-multipliers to this certain class of Banach algebras. Among other the relationship between Arens regularity of the algebra A∗∗ of a weak Arens regular Banach algebra A and the space QMr(A∗) of all bilinear and separately continuous right quasimultipliers of A∗ is investigated. Further, we stablish several properties of the strict and quasi-strict topologies on the QMr(A∗) Mathematics Subject Classification: Primary 47B48; Secondary 46H25

Highlights

The notion of a quasi-multiplier is a generalization of the notion of a multiplier on a Banach algebra and was introduced by Akemann and Pedersen [1] for C∗algebras
McKennon [14] extended the definition to a general complex Banach algebra A with a bounded approximate identity (b.a.i., for brevity) as follows
The aim of this paper is to present a few new statements on quasi-multipliers of the dual A∗ of a Banach algebra A whose second dual has a mixed identity

Summary

Introduction

The notion of a quasi-multiplier is a generalization of the notion of a multiplier on a Banach algebra and was introduced by Akemann and Pedersen [1] for C∗algebras. For a Banach space X, let X∗ be its topological dual. The space A∗∗ equipped with the first (or second) Arens product is a Banach algebra and A is a subalgebra of it. Definition 2.1 A Banach algebra A is called weak Arens regular if for each ξ ∈ A∗ and F, G ∈ A∗∗ we have (F · ξ) · G = F · (ξ · G) Every Arens regular Banach algebra is weak Arens regulsr.

Objectives

Results

Conclusion