I am looking at Murphy's book "$C^*$algebras and operator theory", in the section on abelian Von Neumann algebras (end of chapter 4). There, it is explained that any (unit containing) abelian Von Neumann algebra (say $A$) on a separable Hilbert space (say $H$) with a cyclic vector can be identified via a unitary to an $L^\infty(X,\mu)$ for a second countable compact space $X$.
What I don't understand is the explanation for second countability; namely, using the notations there, why there is a separable $C^*$algebra $B$ contained in $A$ which is strongly dense in $A$? I have looked at the remarks at the beginning of the section, and looked in the book, but I don't see the reason here.
Thanks for any help.