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Since the pioneering work of Dixmier and Segal in the early 50's, the theory of noncommutative LP-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 t-measurability developed in more detail; Jordan *-morphisms (representing quantum measurable transformations) that behave well with regard to t-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 LP 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 LP-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 LP-spaces, but also Orlicz spaces for general von Neumann algebras.

Algebra. --- Lp spaces.

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This tract presents an exposition of methods for testing sets of special functions for completeness and basis properties, mostly in L2 and L2 spaces. The first chapter contains the theoretical background to the subject, largely in a general Hilbert space setting, and theorems in which the structure of Hilbert space is revealed by properties of its bases are dealt with. Later parts of the book deal with methods: for example, the Vitali criterion, together with its generalisations and applications, is discussed in some detail, and there is an introduction to the theory of stability of bases. The last chapter deals with complete sets as eigenfunctions of differential and a table of a wide variety of bases and complete sets of special functions. Dr Higgins' account will be useful to graduate students of mathematics and professional mathematicians, especially Banach spaces. The emphasis on methods of testing and their applications will also interest scientists and engineers engaged in fields such as the sampling theory of signals in electrical engineering and boundary value problems in mathematical physics.

Functions, Special. --- Series, Orthogonal. --- Lp spaces.

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Representations of groups --- Holomorphic functions --- Harmonic functions --- Lp spaces --- Hardy spaces

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Analytical spaces --- Lp spaces. --- Espaces Lp --- Tensor products. --- Produits tensoriels --- Lp spaces --- Tensor products --- Products, Tensor --- Algebras, Linear --- Calculus of tensors --- Tensor algebra --- Spaces, Lp --- Function spaces --- Functional analysis --- Espaces Lp. --- Produits tensoriels.

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Analytical spaces --- Mathématiques --- Wiskunde --- Lp spaces. --- Banach spaces. --- Banach, Espaces de. --- Spaces of measures --- Espaces de mesures --- Spaces of measures. --- Banach, Espaces de --- Espaces de lebesgue