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An infinite dimensional Banach space $X$ is decomposable provided $X$ is the direct sum of two closed infinite dimensional subspaces; equivalently, if there is a bounded linear idempotent operator on $X$ whose rank and corank are both infinite. The first separable indecomposable Banach space was constructed by Gowers and Maurey. It has the stronger property that every infinite dimensional closed subspace is also indecomposable; such a space is said to be HI or hereditarily indecomposable. There do not exist HI Banach spaces having arbitrarily large cardinality (although Argyros did construct non separable HI spaces), but I do not know the answer to:

Question: If the cardinality of a Banach space is sufficiently large, must it be decomposable?

Much is known if $X$ has some special properties (see Zizler's article in volume II of the Handbook of the Geometry of Banach Spaces). Something I observed (probably many others did likewise) around 40 years ago is that the dual to any non separable Banach space is decomposable; I mention it because it is not in Zizler's article (in his discussion of idempotents he is interested in getting more structure--a projectional resolution of the identity) and I did not publish it because it is an easy consequence of lemmas J. Lindenstrauss proved to get projectional resolutions of the identity for reflexive spaces.

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    $\begingroup$ Great question, but I really have no idea about the answer. Last I heard (though I haven't really read the paper, namely arxiv.org/abs/1106.2916 ) was Koszmider's result that it is consistent that there is a space $C(K)$ of density $2^c$, where $c$ is the cardinality of the continuum. In introduction to the preprint, Koszmider mentions that there is a bound on the density of the spaces having the properties that his $C(K)$ has, but that the question posed above seems to still be open. $\endgroup$
    – Philip Brooker
    Commented Aug 8, 2011 at 4:52
  • $\begingroup$ Thanks, Philip. Somehow I missed this recent paper of Koszmider even though I knew his work on indecomposable $C(K)$ spaces. As far as I know, the question does not appear in the published literature. $\endgroup$
    – Bill Johnson
    Commented Aug 8, 2011 at 8:13
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    $\begingroup$ In the paper by Piotr Koszmider; A survey on Banach spaces C(K) with few operators; RACSAM 104 (2), 2010, pp. 309 -326. He mentions this question (Problem 6) and attributes it Argyros; it sounds like the question was communicated to him personally and had not appeared in the literature before this survey. I have a copy of the paper if you want me to send it. You may want to ask Dodos, Lopez-Abad or Todorcevic as well. They have recent work (which appeared in the Advances) on finding a similar bound for spaces containing unconditional basic sequences. $\endgroup$
    – Kevin Beanland
    Commented Aug 9, 2011 at 0:32
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    $\begingroup$ One more related result to point out (for anyone interested) is that for HI spaces the bound is $2^\omega$; since every HI space embedds into $\ell_\infty$ (it seems that many authors independently proved this. The book of Argyros and Todorcevic contains the proof, which is not all that hard.) $\endgroup$
    – Kevin Beanland
    Commented Aug 9, 2011 at 0:35
  • $\begingroup$ Kevin, thanks for mentioning Koszmider's survey paper (which I didn't think to look in for this question). I was sure I'd seen the question published somewhere before and was just about to go looking for it when your comment appeared. $\endgroup$
    – Philip Brooker
    Commented Aug 9, 2011 at 7:21

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According to the recent preprint by Koszmider, Shelah and Świętek under the generalised continuum hypothesis there is no such bound. In particular, one cannot prove the existence of such a bound working merely within the ZFC.

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  • $\begingroup$ Thanks for the reference, Tomek. This is very interesting. The examples are even $C(K)$ spaces! $\endgroup$
    – Bill Johnson
    Commented Mar 8, 2016 at 14:30

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