Sub-Stonean spaces and corona sets
type: Item
instance of: scholarly article
title: Sub-Stonean spaces and corona sets
zbMATH Open document ID: 0539.54029
DOI: 10.1016/0022-1236(84)90028-4
author: Karsten Grove
author: Gert Kjaergård Pedersen
published in: Journal of Functional Analysis
publication date: 1984-01-01T00:00:00Z
review text: A locally compact Hausdorff space X is sub-Stonean if any two disjoint, open \(\sigma\)-compact subsets of X have disjoint, compact closures. Sub- Stonean spaces are, of course, generalizations of Stonean spaces and Rickart spaces, but they are topologically much more intricate. For example, there are connected sub-Stonean spaces of any finite covering dimension, despite the fact that no infinite sub-Stonean space contains any convergent sequences, hence no non-trivial curves. Of particular interest are the corona sets of the form \(\beta(Y)\backslash Y\) for some locally compact, \(\sigma\)-compact space Y, because these sub-Stonean spaces lend themselves to Čech-cohomological considerations. For example, if \(Y=Z\times {\mathbb{R}}_+\) for some compact space Z, and \(X=\beta(Y)\backslash Y,\) then \(H^ 2(Z,{\mathbb{Z}})=0\) if and only if \(H^ 2(X,{\mathbb{Z}})=0\). The theory of sub-Stonean spaces is essential for the solution of the diagonalization problem for \(C(X)\otimes {\mathbb{M}}_ n\) found in the authors' paper ''Diagonalizing matrices over C(X)'' (to appear in the same journal). Non-commutative generalizations of sub-Stonean spaces are found by the second author in \(''SC^*\)-algebras and corona \(C^*\)-algebras: contributions to non-commutative topology'' (to appear in J. Oper. Theory).
Mathematics Subject Classification ID: 54G05
Mathematics Subject Classification ID: 54D35
zbMATH DE Number: 3857961
zbMATH Keywords: locally compact Hausdorff space
zbMATH Keywords: Stonean spaces
zbMATH Keywords: Rickart spaces
zbMATH Keywords: connected sub-Stonean spaces
zbMATH Keywords: finite covering dimension
zbMATH Keywords: corona sets
zbMATH Keywords: locally compact, \(\sigma\)-compact space
zbMATH Keywords: diagonalization problem
MaRDI profile type: Publication
full work available at URL: 0022-1236(84)90028-4
OpenAlex ID: W2087214704
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description: scientific article; zbMATH DE number 3857961