Let $B=\int\_{Y}^{\oplus}B\_yd\_y$ be the direct integral decomposition of vNa $B$ into factors and if $P=\int\_{Y}^{\oplus} p\_ydy$ is a projection in $B$ and $p\_y$ is equivalent to a projection $q\_y$ in $B\_y$ for all or a.e. $y\in Y$. Is there a good choice of $q\_y$ up to equivalence such that $Q=\int\_{Y}^{\oplus} q\_ydy$ makes sense and $P$ and $Q$ are equivalent in $B$?



Disintegration of von Neumann algebra
Suppose B is a trivial bundle whose fibers are type I2 factors and p is a constant section of B corresponding to some projection with 1-dimensional image. Projections with 1-dimensional image in a type I2 factor can be identified with angles, i.e., elements of iR/πiZ. If q is given by some non-measurable section of B, then there is no way to choose its equivalence class in such a way that Q exists.


