Pointwise approximate identities in Banach function algebras

Dales, G. and Ulger, Ali (2020) Pointwise approximate identities in Banach function algebras. Dissertationes Mathematicae (Rozprawy Matematyczne). ISSN 0012-3862 (In Press)

[img]
Text (DUPAIFinalindices)
DUPAIFinalindices.pdf - Accepted Version
Available under License Creative Commons Attribution-NonCommercial.

Download (642kB)

Abstract

In this memoir, we shall study Banach function algebras that have bounded pointwise approximate identities, and especially those that have contractive pointwise approximate identities. ABanach function algebra A is (pointwise) contractive if A and every non-zero, maximal modular ideal in A have contractive (pointwise) approximate identities. Let A be a Banach function algebra with character space Phi_A. We shall show that the existence of a contractive pointwise approximate identity in A depends closely on whether ||varphi|| =1 for each varphi in Phi_A$. The linear span of Phi_A in the dual space A' is denoted by L(A), and this is used to define the BSE norm on A; the algebra A has a BSE norm if this norm is equivalent to the given norm. We shall then introduce and study in some detail the quotient Banach function algebra {mathcal Q}(A)= A''/L(A)^\perp; we shall give various examples, especially uniform algebras and those involving algebras that are standard in abstract harmonic analysis, including Segal algebras with respect to the group algebra of a locally compact group. We shall characterize the Banach function algebrasfor which overline{L(A)}= \ell^{1}(\Phi_A), and then classify contractive and pointwise contractive algebras in the class of unital Banach function algebras that have a BSE norm; they are uniform algebras with specific properties. We shall also give examples of such algebras that do not have a BSE norm. Finally we shall discuss when some classical Banach function algebras of harmonic analysis have non-trivial reflexive closed ideals, and make some remarks on weakly compact homo\-morphisms between Banach function algebras

Item Type:
Journal Article
Journal or Publication Title:
Dissertationes Mathematicae (Rozprawy Matematyczne)
Uncontrolled Keywords:
/dk/atira/pure/subjectarea/asjc/2600
Subjects:
ID Code:
147192
Deposited By:
Deposited On:
08 Sep 2020 15:50
Refereed?:
Yes
Published?:
In Press
Last Modified:
29 Nov 2020 07:02