Pervasive Algebras and Maximal Subalgebras

Gorkin, Pamela and O'Farrell, Anthony G. (2011) Pervasive Algebras and Maximal Subalgebras. Studia Mathematica (206). pp. 1-24. ISSN 0039-3223

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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.

Item Type: Article
Additional Information: 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.
Keywords: Uniform algebra; logmodular algebra; pervasive algebra; maximal subalgebra;
Academic Unit: Faculty of Science and Engineering > Mathematics and Statistics
Item ID: 3766
Depositing User: Prof. Anthony O'Farrell
Date Deposited: 19 Jun 2012 15:36
Journal or Publication Title: Studia Mathematica
Publisher: Polska Akademia Nauk
Refereed: No

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