## Pointwise approximate identities in Banach function algebras

### Volume 557 / 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.