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## Dissertationes Mathematicae

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## Pointwise approximate identities in Banach function algebras

### Volume 557 / 2020

Dissertationes Mathematicae 557 (2020), 1-74 MSC: Primary 46B15; Secondary 46B28, 46B42, 47L10. DOI: 10.4064/dm796-8-2020 Published online: 13 November 2020

#### 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. A Banach 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 $\Vert\,{\cdot}\,\Vert_{\rm BSE}$ 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 algebras for 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 Banach function 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 homomorphisms between Banach function algebras.

#### Authors

• H. G. DalesDepartment of Mathematics and Statistics University of Lancaster
Lancaster, LA1 4YF
United Kingdom
e-mail
• A. ÜlgerDepartment of Mathematics
Boğaziçi University
34450 Bebek, Istanbul
Turkey
e-mail

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