Approximate Identity and Arens Regularity of Some Banach Algebras

2009/04/27 by Kazem Haghnejad Azar, Azar, Kazem Haghnejad, Abdolhamid Riazi +1
Mathematics · #46L06 #46L07 #46L10 #47L25 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #math.FA #math.OA #msc:46L06 #msc:46L07 #msc:46L10 #msc:47L25

paper · pdf · doi:10.48550/arxiv.0904.4102

15 page

arxiv created 2009/04/27 · arxiv updated 2009/12/01

Abstract

Let A be a Banach algebra with the second dual A^**. If A has a bounded approximate identity (=BAI), then A^** is unital if and only if A^** has a weak^* bounded approximate identity(=W^*BAI). If A is Arens regular and A \noindent has a BAI, then A^* factors on both sides. In this paper we introduce new concepts LW^*W and RW^*W- property and we show that under certain conditions if A has LW^*W and RW^*W- property, then A is Arens regular and also if A is Arens regular, then A has LW^*W and RW^*W- property. We also offer some applications of these new concepts for the special algebras l¹(G), L¹(G), M(G), and A(G).

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