### There is no bound on sizes of indecomposable Banach Spaces

by Koszmider and Shelah and Swietek. [KoShSw:1086]

Advances in Math, 2018

Assuming the generalized continuum hypothesis we construct
arbitrarily big indecomposable Banach spaces. i.e., such that
whenever
they are decomposed as X oplus Y, then one of the closed subspaces
X or Y must be finite dimensional. It requires alternative
techniques compared to those which were initiated by Gowers
and
Maurey
or Argyros with the coauthors. This is because hereditarily
indecompo-
sable Banach spaces always embed into ell_infty and so their
density and cardinality is bounded by the continuum and because
dual Banach
spaces of densities bigger than continuum are decomposable by
a result
due to Heinrich and Mankiewicz.
The obtained Banach spaces are of the form C(K) for some compact
connected Hausdorff space and have few operators in the sense
that every
linear bounded operator T on C(K) for every f in C(K) satisfies
T(f)=gf+S(f) where g in C(K) and S is weakly compact or
equivalently strictly singular. In particular, the spaces carry
the
structure of a Banach algebra and in the complex case even the
structure of a C^*-algebra.

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