Pervasive Algebras and Maximal Subalgebras

TL;DR

This paper investigates the properties of maximal and pervasive uniform algebras, exploring their relationships, structural hierarchies, and examples involving various classes of function algebras, with new results connecting pervasiveness and maximality.

Contribution

It introduces new conditions under which pervasive algebras are maximal and constructs examples illustrating the complex hierarchy between these concepts.

Findings

Pervasive and proper algebras with a nonconstant unimodular element contain infinite descending chains.

The lattice of all subsets of N can be embedded into pervasive subalgebras of some C(X).

Strongly logmodular, proper, and pervasive algebras are maximal.

Abstract

A uniform algebra $A$ on its Shilov boundary $X$ is {\em maximal} if $A$ is not $C(X)$ and there is no uniform algebra properly contained between $A$ and $C(X)$. It is {\em essentially pervasive} if $A$ is dense in $C(F)$ whenever $F$ is a proper closed subset of the essential set of $A$. If $A$ is maximal, then it is essentially pervasive and proper. We explore the gap between these two concepts. We show the following: (1) If $A$ is pervasive and proper, and has a nonconstant unimodular element, then $A$ contains an infinite descending chain of pervasive subalgebras on $X$. (2) It is possible to imbed a copy of the lattice of all subsets of $\N$ into the family of pervasive subalgebras of some $C(X)$. (3) In the other direction, if $A$ is strongly logmodular, proper and pervasive, then it is maximal. (4) This fails if the word \lq strongly' is removed. We discuss further examples, involving Dirichlet algebras, $A(U)$ algebras, Douglas algebras, and subalgebras of $H^\infty(\mathbb{D})$. We develop some new results that relate pervasiveness, maximality and relative maximality to support sets of representing measures.