Connection between the predual of a von Neumann algenra and $L^{\infty}(\mu)$ I saw the following conclusion from Miguel Martin and Yoshimichi Ueda's paper.
Let $M$ be an arbitrary von Neuamnn algebra and denote its unique predual by $M_{*}$, and they are known to be non-commutative counterparts to $L_{\infty}(\mu)$ and $L_1(\mu)$, respectively.
How can we view $M_{*}$ as non-commutative counterparts to $L_{\infty}(\mu)$ and $L_1(\mu)$, respectively
 A: The category of commutative von Neumann algebras is
contravariantly equivalent
to the category of compact strictly localizable enhanced measurable spaces
with morphisms being equivalence classes of measurable measure-zero reflecting
maps modulo the equivalence relation of weak equality almost everywhere.
(Here “compact” is an abstract measure-theoretic analogue of Radon measures,
and “strictly localizable” generalizes σ-finiteness.)
Under this correspondence, commutative von Neumann algebras
are precisely algebras of bounded measurable functions
on compact strictly localizable enhanced measurable spaces,
modulo equality almost everywhere.
Preduals of commutative von Neumann algebras
are precisely the vector spaces of finite complex-valued measures on such spaces,
or, equivalently, L^1-spaces with respect to any faithful measure.
Thus, it is completely natural to see the predual of a noncommutative
von Neumann algebra as the vector space of measures
on a noncommutative measurable space.
