Trying to complete exercise in Arveson's book. I've showed that unit ball in $\mathcal B (H)$ (bounded linear operators on Hilbert space ) is compact w.r.t. WOT. For a separable Hilbert space, I've showed that unit ball of $\mathcal B(H)$ is homeomorphic to a compact metric space for the weak operator topology.
The next exercise was that every von Neumann algebra $\mathcal M$ acting on separable $H$ contains a unital $C^*$ algebra that is separable (contains norm-dense subset) and weakly dense in $\mathcal M$. For this one I've took the unit ball of $\mathcal M$ and unit ball in $\mathcal B (H)$ is separable since it is a compact metric space, the unit ball of $\mathcal M$ is separable hence there exists countable dense subset of it (but not in norm). By adjoining the identity of the $\mathcal M$ we can make that dense subset unital I've considered the $C^*$-algebra generated by that dense subset and unit of $\mathcal M$. Since in any Banach space $\cup_{n \in \mathbb {N}} nB_1(0)$ is dense in the space if it is separable, this $C^*$ algebra is weakly dense in $\mathcal M$. However I still couldn't find a separable one, can someone help?
Best Answer
So you have a countable subset $S$ of the unit ball of $\mathcal M$ that is weakly dense in the unit ball of $\mathcal M$. You correctly argue that the C* algebra generated by $S$ is weakly dense in $\mathcal M$. This C* algebra is also separable:
Since $S$ is countable it spans a vector space of countable algebraic dimension. Let $\{ x_n \mid n\in\Bbb N\}$ be a basis of this space. Then the set consisting of finitely many products of the $x_n$ and $x_n^*$ is still countable. The complex span of those products then remains a vector space of countable algebraic dimension (hence is separable wrt any norm on it) - but now it is also closed under products and the $*$-operation. Its norm closure will then also be separable and closed under products and the $*$-operation, ie its norm closure will be a separable sub-algebra of $\mathcal M$. In fact the construction thus perfomed is exactly the C* algebra generated by $S$.