Abelian von Neumann Algebras on non-separable Hilbert spaces

Question

Is there a classification of Abelian von Neumann algebras on non-separable Hilbert spaces? For a classification of Abelian von Neumann algebras on separable Hilbert spaces, see this link.

Answer 1

Every commutative von Neumann algebra $A$ is $^*$-isomorphic to $L_\infty(\Omega,\mu)$ for some locally compact Hausdorff space and positive measure $\mu$.

Idea of the proof is the following. Decompose $$ A\cong\coprod\limits_{\nu\in\Lambda} A_\nu $$ into the coproduct of commutative von Neumann algebras with cyclic vector. For every commutative von Neumann algebra $A_\nu$ with cyclic vector there exist compact Hausdorff space $\Omega_\nu$ and regular bounded Borel measure $\mu_\nu$, such that $A_\nu$ is $^*$-isomorphic to $L_\infty(\Omega_\nu,\mu_\nu)$. Then one can show that $$ A\cong L_\infty\left(\coprod\limits_{\nu\in\Lambda} \Omega_\nu,\coprod\limits_{\nu\in\Lambda} \mu_\nu\right) $$

See Theory of operator algebras Part I by M. Takesaki, chapter 3, theorem 1.18.