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Is there any criterion answering the question:

Let $E$ be a Banach space. When does the Banach space $\mathcal{B}(E)$ of all bounded operators on $E$ contain a copy of $\ell^\infty(\Gamma)$? Here $\Gamma$ is an arbitrary index set, perhaps uncountable.

Of course, the answer is easy when $E$ is a Hilbert space with density character equal to the cardinality of $\Gamma$.

Thank you.

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A sufficient condition would be the existence of a complemented subspace with an unconditional basis. On that subspace, then, you can just copy the argument that works for a Hilbert space. I don't know whether that is necessary, though. I think the Argyros-Haydon space has a separable space of bounded linear operators, so some supplementary information is necessary. –  t.b. Oct 8 '11 at 10:30
    
Of course, but I am pretty sure that some time ago I've seen a paper precisely on that matter. Unfortunately, browsing mathscinet gives me nothing... –  Sellapan Nathan Oct 8 '11 at 10:41
    
Was it by any chance this paper? Here's its MathSciNet review. –  t.b. Oct 8 '11 at 10:45
    
You are superb! Thank you, thanks a lot! –  Sellapan Nathan Oct 8 '11 at 10:48
    
You're welcome :) I didn't know it, I just Googled for embedding l^\infty into bounded operators and it was one of the first hits. –  t.b. Oct 8 '11 at 10:52

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