Let $A$ be a separable $\mathrm{C}^*$-algebra and let $\pi_U$ be its universal representation. Denote by $M=\pi_U(A)''$ the universal enveloping von Neumann algebra of $A$ (which is isomorphic to $A^{**}$). I have the following questions:

1) Is strong operator topology of $M$ metrizable, when restricted to the closed unit ball?

2) Let $a$ be an element of the closed unit ball of $M$. Thanks to Kaplansky theorem we can find a net $(a_i)_{i\in I}$ of elements from the closed unit ball of $A$, which converge to $a$ in strong operator topology. Can we find a sequence which converges to $a$?

3) Is $M$ $\sigma$-finite? (Von Neumann algebra is $\sigma$-finite if any collection $\{p_i\}_{i\in I}$ of pairwise orthogonal nonzero projections is at most countable).

If this help, one can change strong operator topology in questions 1), 2) to weak operator topology.

I suspect in general these statements are not true, however only concrete cases that I am aware of are $K(\ell^2)^{**}=B(\ell^2)$ and $c_0^{**}=\ell^{\infty}$, and these von Neumann algebras can be represented on separable Hilbert spaces.


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