Sub-Stonean spaces and corona sets
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).
locally compact, \(\sigma\)-compact space
scientific article; zbMATH DE number 3857961
scientific article