Gorkin, Pamela and O'Farrell, Anthony G.
Pervasive Algebras and Maximal Subalgebras.
Studia Mathematica (206).
A uniform algebra A on its Shilov boundary X is maximal if
A is not C(X) and there is no uniform algebra properly 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 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 ‘strongly’ is removed.
We discuss further examples, involving Dirichlet algebras, A(U)
algebras, Douglas algebras, and subalgebras of H1(D). We develop
some new results that relate pervasiveness, maximality and relative
maximality to support sets of representing measures.
||Preprint version of original published article. The definitive version of the article is available at Studia Mathematica No.206(2011) pp.1-24; doi:10.4064/sm206-1-1 . The second author was partially-supported by the HCAA network.
Part of this work was done at the meeting on Banach Algebras held at
Bedlewo in July 2009. The support for the meeting by the Polish Academy
of Sciences, the European Science Foundation under the ESF-EMS-ERCOM
partnership, and the Faculty of Mathematics and Computer Science of the
Adam Mickiewicz University at Poznan is gratefully acknowledged. The first
author is grateful to the London Mathematical Society for travel funding.
||Uniform algebra; logmodular algebra; pervasive
algebra; maximal subalgebra;
||Science & Engineering > Mathematics & Statistics
Prof. Anthony O'Farrell
||19 Jun 2012 15:36
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