Pervasive algebras and maximal subalgebras

• Autorzy: Pamela Gorkin , Anthony G. O'Farrell
• Języki publikacji: 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.

Kategorie tematyczne

• 46J10: Banach algebras of continuous functions, function algebras

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2011
• Tom: 206
• Numer: 1
• Strony: 1-24

Daty

wydano

2011

Twórcy

autor

Pamela Gorkin

• Department of Mathematics, Bucknell University, Lewisburg, PA 17837, U.S.A.

autor

Anthony G. O'Farrell

• Department of Mathematics, National University of Ireland, Maynooth, Maynooth, Co. Kildare, Ireland

Identyfikatory

DOI

10.4064/sm206-1-1