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Studia Mathematica

2011 | 206 | 1 | 1-24
Tytuł artykułu

Pervasive algebras and maximal subalgebras

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EN
Abstrakty
EN
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 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 $H^{∞}(𝔻)$, and develop new results that relate pervasiveness, maximality, and relative maximality to support sets of representing measures.
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Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
1-24
Opis fizyczny
Daty
wydano
2011
Twórcy
autor
• Department of Mathematics, Bucknell University, Lewisburg, PA 17837, U.S.A.
autor
• Department of Mathematics, National University of Ireland, Maynooth, Maynooth, Co. Kildare, Ireland
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