Weight theory for ultraproducts

Abstract

For a family of von Neumann algebras $\mathcal{M}_j$ equipped with normal weights $\varphi_j$ we define the ultraproduct weight $(\varphi_j)_\omega$ on the Groh--Raynaud ultrapower $\prod_{j, \omega} \mathcal{M}_j$. We prove results about Tomita-Takesaki modular theory and consider ultraproducts of spatial derivatives. This extends results by Ando--Haagerup and Raynaud for the state case. We give some applications to noncommutative $L^p$-spaces and indicate how ultraproducts of weights appear naturally in transference results for Schur and Fourier multipliers. Using ideas from complex interpolation with respect to ultraproduct weights, we give a new proof of a theorem by Raynaud which shows that $\prod_{j, \omega} L^p(\mathcal{M}_j) \simeq L^p(\prod_{j, \omega} \mathcal{M}_j )$. We complement the paper by showing that spatial derivatives take a natural form in terms of noncommutative $L^p$-spaces.