The Kadison-Singer problem is considered in relation to the separable Hilbert space:

KS: Does every pure state on the diagonal (atomic) masa of $B(\ell_2)$ has a unique extension to $B(\ell_2)$?

What is the status of this problem for non-separable Hilbert spaces? Most of operator-algebraits are uninterested in (starting a day with) non-separable Hilbert spaces, so probably they don't care. But if the (classical) KS problem has a negative solution, this solution *must* depend on the underlying ultrafilter $p\in \beta \mathbb{N}\setminus \mathbb{N}$ (it cannot be, for example, rare).

Sometimes weird ultrafilter over uncountable sets are (consistently) easier to grasp. This indicates that perhaps one should look at $\ell_2(\kappa)$ for some $\kappa$ big enough.