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Pointwise approximate identities in Banach function algebras

Research output: Contribution to journalJournal article

Forthcoming
<mark>Journal publication date</mark>10/08/2020
<mark>Journal</mark>Dissertationes Mathematicae (Rozprawy Matematyczne)
Number of pages74
Publication StatusAccepted/In press
<mark>Original language</mark>English

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