Notes on noncommutative LP and Orlicz spaces

Since the pioneering work of Dixmier and Segal in the early 50’s, the theory of noncommutative L^P-spaces has grown into a very refined and important theory with wide applications. Despite this fact there is as yet no self-contained peer-reviewed introduction to the most general version of this theory in print. The present work aims to fill this vacuum, in the process giving fresh impetus to the theory. The first part of the book presents: the introductory theory of von Neumann algebras – also including the slightly less common theory of generalized positive operators; the various notions of measurability, allowing the interpretation of unbounded affiliated operators as “quantum”" measurable functions, with the crucial notion of τ-measurability developed in more detail; Jordan *-morphisms (representing quantum measurable transformations) that behave well with regard to τ-measurability; and finally the different types of weights that occur naturally in the theory, before presenting a Radon-Nikodym theorem for such weights. The core, second part of the book is devoted to first developing the noncommutative theory of decreasing rearrangements, before using that technology to present the basic theory of L^P and Orlicz spaces for semifinite algebras, and then the notion of crossed product, as well as the technology underlying it, indispensable for the theory of Haagerup L^P-spaces for general von Neumann algebras. With this as a foundation, we are then finally ready to present the basic structural theory of not only Haagerup L^P-spaces, but also Orlicz spaces for general von Neumann algebras.