Sh:1086

• Koszmider, P., Shelah, S., & Świȩtek, M. (2018). There is no bound on sizes of indecomposable Banach spaces. Adv. Math., 323, 745–783. arXiv: 1603.01753 DOI: 10.1016/j.aim.2017.11.002 MR: 3725890
• 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 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.
• Version 2017-11-04_11 (31p) published version (39p)

@article{Sh:1086,
 author = {Koszmider, Piotr and Shelah, Saharon and {\'S}wi{\c{e}}tek, Micha{\l}},
 title = {{There is no bound on sizes of indecomposable Banach spaces}},
 journal = {Adv. Math.},
 fjournal = {Advances in Mathematics},
 volume = {323},
 year = {2018},
 pages = {745--783},
 issn = {0001-8708},
 mrnumber = {3725890},
 mrclass = {46B25 (03E35 06E20 46B28 47B38 47L10 54D30)},
 doi = {10.1016/j.aim.2017.11.002},
 note = {\href{https://arxiv.org/abs/1603.01753}{arXiv: 1603.01753}},
 arxiv_number = {1603.01753}
}