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Separation and Ultraseparation Properties for Continuous Function Spaces
Author(s) -
Ellis A. J.
Publication year - 1984
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-29.3.521
Subject(s) - hausdorff space , mathematics , disjoint sets , intersection (aeronautics) , pure mathematics , banach space , paracompact space , function (biology) , interpolation (computer graphics) , function space , continuous function (set theory) , ideal (ethics) , space (punctuation) , interpolation space , topological space , normal space , discrete mathematics , functional analysis , topological vector space , computer science , image (mathematics) , biology , evolutionary biology , gene , philosophy , artificial intelligence , aerospace engineering , chemistry , engineering , operating system , biochemistry , epistemology
The objects of study are Banach spaces L of continuous real‐valued functions on a compact Hausdorff space Ω, containing constants and separating points. If the intersection of an arbitrary family of M ‐ideals in L is an M ‐ideal then, in certain specified circumstances, it is shown that L is necessarily equal to C R (Ω); these results extend a well‐known theorem of Gleit [ 9 ]. An intrinsic characterisation of ultraseparation for L is obtained, involving standard separation properties for closed disjoint subsets of Ω by functions in L ; an application to interpolation sets for uniform algebras is given. Certain general classes of function spaces are shown to be ultraseparating relative to their Silov boundaries.