Pervasive algebras and maximal subalgebras

Pamela Gorkin¹ ; Anthony G. O'Farrell²

¹ Department of Mathematics Bucknell University Lewisburg, PA 17837, U.S.A.
² Department of Mathematics National University of Ireland, Maynooth Maynooth, Co. Kildare, Ireland

Studia Mathematica, Tome 206 (2011) no. 1, pp. 1-24

Résumé

A uniform algebra $A$ on its Shilov boundary $X$ is maximal if $A$ is not $C(X)$ and no uniform algebra is strictly contained between $A$ and $C(X)$. It is 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: (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 find a compact Hausdorff space $X$ such that there is an isomorphic copy of the lattice of all subsets of $\def\N{\mathbb N}\N$ in the family of pervasive subalgebras of $C(X)$. (3) In the other direction, if $A$ is strongly logmodular, proper and pervasive, then it is maximal. (4) This fails if the word “strongly” is removed. We discuss examples involving Dirichlet algebras, $A(U)$ algebras, Douglas algebras, and subalgebras of $H^\infty(\mathbb{D})$, and develop new results that relate pervasiveness, maximality, and relative maximality to support sets of representing measures.

Détail

DOI : 10.4064/sm206-1-1

Mots-clés : uniform algebra its shilov boundary maximal uniform algebra strictly contained between essentially pervasive dense whenever proper closed subset essential set maximal essentially pervasive proper explore gap between these concepts pervasive proper has nonconstant unimodular element contains infinite descending chain pervasive subalgebras possible compact hausdorff space there isomorphic copy lattice subsets def mathbb family pervasive subalgebras other direction strongly logmodular proper pervasive maximal fails word strongly removed discuss examples involving dirichlet algebras algebras douglas algebras subalgebras infty mathbb develop results relate pervasiveness maximality relative maximality support sets representing measures