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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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    It would be helpful for readers to include this in the body of the question.2011-10-08

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