Measurable Bundles of Noncommutative $L_p$-Spaces Associated with a Center-valued Trace

Suppose that $M$ is a finite von Neumann algebra, $\Phi$ is a faithful normal trace on $M$ with values in the center of $M$, $L_p(M,\Phi)$ is the Banach–Kantorovich space of all measurable operators associated with $M$ and $p$-integrable with respect to $\Phi$, $p\ge 1$. We give a representation of $L_p(M,\Phi)$ as a measurable bundle of noncomutative $L_p$-spaces associated with number traces. We also prove a “pasting” theorem for noncommutative $L_p$-spaces.