Lecture Notes on Noncommutative Lp-Spaces

Abstract

These lecture notes were written during a mini-course on noncommutative Lp-spaces at the Basque Center of Applied Mathematics. It starts presenting the theory of weights and traces in von Neumann algebra, followed by the theory of noncommutative measure. After that, it presents a construction of these spaces in all detail and give some important results e.g. the noncommutative Radon-Nikodym theorem.

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• Itô's formula for noncommutative $C^2$ functions of free Itô processes
• Multiple operator integrals in non-separable von Neumann algebras
• Higher derivatives of operator functions in ideals of von Neumann algebras
• Perturbations of KMS States and Noncommutative Lp-spaces
• Modular Structures on Trace Class Operators and Applications to Themodynamical Equilibrium States of Infinitely Degenerate Systems