Let $M$ be a von Neumann algebra which acts faithfully on a Hilbert space of density character $\kappa$ but does not on a Hilbert space of density character $\lambda<\kappa$ (that is, the density character of the predual $M_0$ is $\kappa$. Does $M$ contain a subalgebra *-isomorphic to $\ell^\infty(\kappa)$? Does $M_0$ contain a complemented subspace isomorphic as a Banach space to $\ell_1(\kappa)$?

A density character is the minimal cardinality of a dense set.