Pervasive algebras and maximal subalgebras
Tom 206 / 2011
Streszczenie
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.
