Suppose $M$ is a Von Neumann Algebra. (VNA) For me, these are subsets of some $B(H)$ that are $*$-algebras, containing the $1$ of $B(H)$, that are Weak Operator (WO) closed, or equivalently Strong Operator (SO) closed.
For a given subset $S$ of $M$ one can always refer to the VNA generated relative to $M$ by $S$. This is the smallest VNA containing $S$ that is contained in $M$, or equivalently it's an intersection over all the VNAs containing $S$ that are contained in $M$. Is there some constructive expression of this, as there is in so many other cases of similar concepts? I don't think the weak operator closure of the noncommuting-variable-polynomials evaluated at the elements of $S$ is sufficient, because operator multiplication is not continuous from the product of two WO topologies to the WO topology. (I only know that I cannot prove that it is. Does someone have an example of when this pathology arises?)
Anyway to "get my hands on" $\langle S\rangle$ would be appreciated, thanks!