Pervasive algebras and maximal subalgebras

Pamela Gorkin, and Anthony G. O'Farrell. "Pervasive algebras and maximal subalgebras." Studia Mathematica 206.1 (2011): 1-24. <http://eudml.org/doc/285555>.

@article{PamelaGorkin2011,
abstract = {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 $H^\{∞\}()$, and develop new results that relate pervasiveness, maximality, and relative maximality to support sets of representing measures.},
author = {Pamela Gorkin, Anthony G. O'Farrell},
journal = {Studia Mathematica},
keywords = {uniform algebra; logmodular algebra; pervasive algebra; maximal subalgebra},
language = {eng},
number = {1},
pages = {1-24},
title = {Pervasive algebras and maximal subalgebras},
url = {http://eudml.org/doc/285555},
volume = {206},
year = {2011},
}

TY - JOUR
AU - Pamela Gorkin
AU - Anthony G. O'Farrell
TI - Pervasive algebras and maximal subalgebras
JO - Studia Mathematica
PY - 2011
VL - 206
IS - 1
SP - 1
EP - 24
AB - 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 $H^{∞}()$, and develop new results that relate pervasiveness, maximality, and relative maximality to support sets of representing measures.
LA - eng
KW - uniform algebra; logmodular algebra; pervasive algebra; maximal subalgebra
UR - http://eudml.org/doc/285555
ER -