# Universal enveloping von Neumann algebra of a separable $C^*$ algebra

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.