Weight theory for ultraproducts

2016/05/24 by Martijn Caspers, Caspers, Martijn · 1 citation
Mathematics · #Advanced Operator Algebra Research #Random Matrices and Applications #Advanced Algebra and Geometry

paper · pdf · doi:10.48550/arxiv.1605.07435

Abstract

For a family of von Neumann algebras M_j equipped with normal weights φ_j we define the ultraproduct weight (φ_j)_ω on the Groh--Raynaud ultrapower ∏_{j, ω} 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 ∏_{j, ω} L^p(M_j) ≃ L^p(∏_{j, ω} M_j ). We complement the paper by showing that spatial derivatives take a natural form in terms of noncommutative L^p-spaces.

Cited by

• Model Theory of General von Neumann Algebras I: Generalized Ocneanu Ultraproducts

Related

• Ultraproducts of von Neumann algebras
• Ultraproducts of crossed product von Neumann algebras
• Ultraproducts of crossed product C*algebras
• Ultraproducts, QWEP von Neumann Algebras, and the Effros-Maréchal Topology
• Model Theory of General von Neumann Algebras I: Generalized Ocneanu Ultraproducts