Radon-Nikodym property of $B(\mathcal{H})$

Question

Let $\mathcal{H}$ be a separable Hilbert space and $B(\mathcal{H})$ the space of bounded linear operators mapping $\mathcal{H}$ to $\mathcal{H}$. Does $B(\mathcal{H})$ have the Radon-Nikodym property? If so, I would be grateful for a reference.

Thank you!

Answer 1

No. Banach space $l^\infty$ is isomorphic to a subspace of $B(\mathcal H)$ (operators diagonal for a certain orthonormal basis). But $l^\infty$ fails the RNP, thus $B(\mathcal H)$ fails the RNP.