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

Piotr Koszmider

Studia Mathematica, Tome 209 (2012), p. 95-117 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $K₊^{\prime } = K₊∖K_{1,0}$ and $K₊^{\prime \prime } = K₊∖(K_{1,0} ∪ K_{2,0})$.

Détail

Publié le : 2012-01-01

Zbl 1272.46004

EUDML-ID : urn:eudml:doc:285470