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

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.

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.