I am reading Arveson's Notes On Extensions of $C^*$-algebras, in proving the Corollary to Thm2, he seems to assume that $\mathcal{K}(\mathcal{H})$, the space of compact operators on a separable Hilbert space, is separable as a topological space, i.e., it contains a countable dense set.
This is a little bit surprising to me. If we just take $\mathcal{H}=\ell^2$ and $F$ defined by:\begin{equation} F(\sum\alpha_n e_n)=(\sum\alpha_n f_n)e_1, \end{equation}
where $\{e_n\}$ is the canonical basis for $\ell^2$. That is, $F$ maps everything to the first coordinate, $F e_n=f_n e_1$. Since $(f_n)$ can be be arbitrary sequences in $\ell^{\infty}$, which is a non-separable space.
My argument seems to imply that even the space of rank-one operators is not separable.
Where did I make a mistake? Thanks!