Property $P$ and direct integral decomposition of $W^*$-algebras

Abstract:

If $\mathcal {A}$ is a $W{{\text {-}}^\ast }$ algebra on separable Hilbert space H, and if $\mathcal {A}(\lambda )$ are the factors in the direct integral decomposition of $\mathcal {A}$, then $\mathcal {P} = \{ \lambda |\mathcal {A}(\lambda )$ has property P} is $\mu$-measurable, and $\mathcal {A}$, has property P iff $\mathcal {A}(\lambda )$ has property P $\mu$-a.e.