Property $L$ and direct integral decompositions of $W-\ast$algebras

Abstract:

If $\mathcal {A}$ is a type $\mathrm {II}_\infty W\text {-}*$ algebra on separable Hilbert space $H,\mathcal {A}$ is spatially isomorphic to $\mathfrak {B} \otimes B(K),\mathfrak {B}$ of type $\mathrm {II}_1$, K a separable Hilbert space. If $\mathcal {A}(\lambda )$ are the factors in the direct integral decomposition of $\mathcal {A}$, the set $\mathfrak {L} = \{ \lambda |\mathcal {A}(\lambda )$ has property L} is $\mu$-measurable, and $\mathcal {A}$ has property L iff $\mu (\Lambda - \mathfrak {L}) = 0$.