A C(K) Banach space which does not have the Schroeder-Bernstein property

• Autorzy: Piotr Koszmider
• Języki publikacji: EN

Abstrakty

EN

We construct a totally disconnected compact Hausdorff space K₊ which has clopen subsets K₊'' ⊆ K₊' ⊆ K₊ such that K₊'' is homeomorphic to K₊ and hence C(K₊'') is isometric as a Banach space to C(K₊) but C(K₊') is not isomorphic to C(K₊). This gives two nonisomorphic Banach spaces (necessarily nonseparable) of the form C(K) which are isomorphic to complemented subspaces of each other (even in the above strong isometric sense), providing a solution to the Schroeder-Bernstein problem for Banach spaces of the form C(K). The subset K₊ is obtained as a particular compactification of the pairwise disjoint union of an appropriately chosen sequence $(K_{1,n} ∪ K_{2,n})_{n ∈ ℕ}$ of Ks for which C(K)s have few operators. We have $K₊' = K₊∖K_{1,0}$ and $K₊'' = K₊∖(K_{1,0} ∪ K_{2,0})$.

Kategorie tematyczne

• 46E15: Banach spaces of continuous, differentiable or analytic functions
• 06E15: Stone spaces (Boolean spaces) and related structures
• 54G20: Counterexamples
• 46B03: Isomorphic theory (including renorming) of Banach spaces

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2012
• Tom: 212
• Numer: 2
• Strony: 95-117

Daty

wydano

2012

Twórcy

autor

Piotr Koszmider

• Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, P.O. Box 21, 00-956 Warszawa, Poland
• Instytut Matematyki Politechniki Łódzkiej, Wólczańska 215, 90-924 Łódź, Poland
• Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain

Identyfikatory

DOI

10.4064/sm212-2-1