SUBSPACES WITH NONINVERTIBLE ELEMENTS IN ReC(X)

Let X be a compact Hausdorff space, and let M be a subspace of ReC(X) consisting only of noninvertible elements. We show that there exist closed sets Y ⊂ X such that each element of M has a zero in Y and no closed subset of Y has this property; furthermore, such a Y is a singleton, or has no isolate...

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Bibliographic Details
Published in:The Rocky Mountain journal of mathematics 2005-01, Vol.35 (5), p.1513-1521
Main Author: HAGHIGHI, M.H. SHIRDARREH
Format: Article
Language:English
Subjects:
Algebra
Mathematical theorems
Scalars
Online Access:Get full text
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Summary:Let X be a compact Hausdorff space, and let M be a subspace of ReC(X) consisting only of noninvertible elements. We show that there exist closed sets Y ⊂ X such that each element of M has a zero in Y and no closed subset of Y has this property; furthermore, such a Y is a singleton, or has no isolated points. If M has finite codimension n and Y is not a singleton, then Y is a union of at most n nontrivial connected components. We also show that positive functionals exist in M⊥.
ISSN:0035-7596
1945-3795
DOI:10.1216/rmjm/1181069648