Ultrapowers of spectral subspaces

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