Pervasive Algebras and Maximal Subalgebras

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

Thin Sequences and Their Role in HpH^p Theory, Model Spaces, and Uniform Algebras

In this paper we revisit some facts about thin interpolating sequences in the unit disc from three perspectives: uniform algebras, model spaces, and $H^p$ spaces. We extend the notion of asymptotic interpolation to $H^p$ spaces, for $1 \leq p \leq \infty$, providing several new ways to think about these sequences.Comment: v1: 21 pages; To appear in Rev. Mat. Iberoa

The group of invariants of an inner function with finite spectrum

This paper determines the group of continuous invariants corresponding to an inner function $\Theta$ with finitely many singularities on the unit circle $T$; that is, the continuous mappings $g: T \to T$ such that $\Theta \circ g = \Theta$ on \T. These mappings form a group under composition.Comment: 14 pages, 3 figures, submitted for publication May 201