Weight theory for ultraproducts

Abstract

For a family of von Neumann algebras ${\mathcal{M}}_j$ equipped with normal weights ${\phi }_j$ we define the ultraproduct weight $({\phi }_j{)}_{\omega }$ 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.

32 pages