Suppose $\Omega$ is a $\sigma$-finite measure space with measure $\mu.$ Let $\mathcal M\subseteq B(H)$ be a von Neumann algebra.


1. Can an element of $L\_\infty(\Omega)\overline{\otimes}\mathcal M$ be regarded as a measurable map from $\Omega$ to $\mathcal M$? A reference where the properties of $L\_\infty(\Omega)\overline{\otimes}\mathcal M$ have been studied will be appreciated also.
2. Is it true that there exists a canonical injective $\*$-homomorphism $\pi:L\_\infty(\Omega;\mathcal M)\to L\_\infty(\Omega)\overline{\otimes}\mathcal M$ such that $\pi(L\_\infty(\Omega;\mathcal M)) $ is dense in $L\_\infty(\Omega)\overline{\otimes}\mathcal M$ in weak operator topology? In above $L\_\infty(\Omega;\mathcal M)$ is all strongly measurable essentially bounded $\mathcal M$-valued functions.



Semi-commutative von Neumann algebras
Assuming $H$ is separable, $L^\infty(\Omega)\overline{\otimes} \mathcal{M}$ can be identified with the essentially bounded weakly measurable functions from $\Omega$ into $\mathcal{M}$. (Weakly measurable = its composition with any normal state on $\mathcal{M}$ is measurable.) This is a minor variation on Theorem 6.5.8 of my book *Mathematical Quantization*. Here the $\mathcal{M}\_x$ appearing in that theorem aren't factors, but they all equal $\mathcal{M}$.


