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.