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

Piotr Koszmider. "A C(K) Banach space which does not have the Schroeder-Bernstein property." Studia Mathematica 212.2 (2012): 95-117. <http://eudml.org/doc/285470>.

@article{PiotrKoszmider2012,
abstract = {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 $K₊^\{\prime \prime \} = K₊∖(K_\{1,0\} ∪ K_\{2,0\})$.},
author = {Piotr Koszmider},
journal = {Studia Mathematica},
keywords = {Banach spaces of continuous functions; Stone spaces; Boolean algebras; Schroeder-Bernstein property},
language = {eng},
number = {2},
pages = {95-117},
title = {A C(K) Banach space which does not have the Schroeder-Bernstein property},
url = {http://eudml.org/doc/285470},
volume = {212},
year = {2012},
}

TY - JOUR
AU - Piotr Koszmider
TI - A C(K) Banach space which does not have the Schroeder-Bernstein property
JO - Studia Mathematica
PY - 2012
VL - 212
IS - 2
SP - 95
EP - 117
AB - 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 $K₊^{\prime \prime } = K₊∖(K_{1,0} ∪ K_{2,0})$.
LA - eng
KW - Banach spaces of continuous functions; Stone spaces; Boolean algebras; Schroeder-Bernstein property
UR - http://eudml.org/doc/285470
ER -