There is no bound on sizes of indecomposable Banach spaces

Abstract

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 indecomposable 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.