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arXiv · 2605.21873

Ultrapowers of spectral subspaces

Hiroshi Ando·Isaac Goldbring
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Abstract

We prove, for any W$^*$-probability space $(M,\varphi)$ 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 $M(\sigma^\varphi,F)^{\mathcal{U}}$ of the spectral subspace $M(\sigma^\varphi,F)$ is a proper subset of the spectral subspace $M^{\mathcal{U}}(\sigma^{\varphi^{\mathcal{U}}},F)$. We discuss the model-theoretic implications of this result.

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Hiroshi Ando, Isaac Goldbring. 2026-05-21. Ultrapowers of spectral subspaces. https://arxiv.org/abs/2605.21873

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