Ultrapowers of spectral subspaces

Abstract

We prove, for any W${}^{\ast }$-probability space $(M,\ddot{I}†)$ where $M$ is a type ${\mathrm{III}}_1$ factor, any nontrivial, proper closed $F\subseteq \mathbb{R}$, and any nonprincipal ultrafilter $\mathcal{U}$ on $\mathbb{N}$, that the ultrapower $M(\ddot{I}{ƒ}^{\ddot{I}}†,F{)}^{\mathcal{U}}$ of the spectral subspace $M(\ddot{I}{ƒ}^{\ddot{I}}†,F)$ is a proper subset of the spectral subspace $M^{\mathcal{U}}(\ddot{I}{ƒ}^{\ddot{I}{†}^{\mathcal{U}}},F)$. We discuss the model-theoretic implications of this result.

10 pages; first draft; comments welcome!