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.