Ultraproducts of factorial $W^*$-bundles

Abstract

This paper investigates factorial $W^*$-bundles and their ultraproducts. More precisely, a $W^*$-bundle is factorial if the von Neumann algebras associated to its fibers are all factors. Let $M$ be the tracial ultraproduct of a family of factorial $W^*$-bundles over compact Hausdorff spaces with finite, uniformly bounded covering dimensions. We prove that in this case the set of limit traces in $M$ is weak$^*$-dense in the trace space $T(M)$. This in particular entails that $M$ is factorial. We also provide, on the other hand, an example of ultraproduct of factorial $W^*$-bundles which is not factorial. Finally, we obtain some results of model-theoretic nature: if $A$ and $B$ are exact, $\mathcal{Z}$-stable $C^*$-algebras, or if they both have strict comparison, then $A \equiv B$ implies that $T(A)$ is Bauer if and only if $T(B)$ is. If moreover both $T(A)$ and $T(B)$ are Bauer simplices and second countable, then the sets of extreme traces $\partial_e T(A)$ and $\partial_e T(B)$ have the same covering dimension.