TITLE: Separable von Neumann algebra
QUESTION [11 upvotes]: What is the simplest argument which shows that each infinite dimensional von Neumann algebra is not separable (in the norm topology)? It seems that this is a kind of folklore: at least I never saw the proof of this fact.

REPLY [4 votes]: Suppose that we have an infinite-dimensional von Neumann algebra. In particular, this is a C*-algebra so it contains an infinite-dimensional abelian C*-algebra $A$. Consequently, $\overline{A}^{{\rm WOT}}$ is a non-separable abelian von Neumann algebra. No need to invoke the fact that it is isomorphic to $L_\infty(\mu)$ for some measure $\mu$, simply use the Borel functional calculus to build an uncountable, closed discrete subset in $\overline{A}^{{\rm WOT}}$ just like you would do in $L_\infty$.
The fact that every infinite-dimensional C*-algebra contains an infinite-dimensional abelian subalgebra is fairly elementary and can be found in Kadison-Ringrose.