The theme of the first Abel Symposium was operator algebras in a wide sense. In the last 40 years operator algebras have developed from a rather special discipline within functional analysis to become a central field in mathematics often described as "non-commutative geometry." It has branched out in several sub-disciplines and made contact with other subjects. The contributions to this volume give a state-of-the-art account of some of these sub-disciplines and the variety of topics reflect to some extent how the subject has developed. This is the first volume in a prestigious new book series linked to the Abel prize.

This is the first of two volumes presenting the theory of operator algebras with applications to quantum statistical mechanics. The authors' approach to the operator theory is to a large extent governed by the dictates of the physical applications. The book is self-contained and most proofs are presented in detail, which makes it a useful text for students with a knowledge of basic functional analysis. The introductory chapter surveys the history and justification of algebraic techniques in statistical physics and outlines the applications that have been made.The second edition contains new and improved results. The principal changes include: A more comprehensive discussion of dissipative operators and analytic elements; the positive resolution of the question of whether maximal orthogonal probability measure on the state space of C-algebra were automatically maximal along all the probability measures on the space.

This book combining wavelets and the world of the spectrum focuses on recent developments in wavelet theory, emphasizing fundamental and relatively timeless techniques that have a geometric and spectral-theoretic flavor. The exposition is clearly motivated and unfolds systematically, aided by numerous graphics.This self-contained book deals with important applications to signal processing, communications engineering, computer graphics algorithms, qubit algorithms and chaos theory, and is aimed at a broad readership of graduate students, practitioners, and researchers in applied mathematics and engineering. The book is also useful for other mathematicians with an interest in the interface between mathematics and communication theory.

For almost two decades this has been the classical textbook on applications of operator algebra theory to quantum statistical physics. It describes the general structure of equilibrium states, the KMS-condition and stability, quantum spin systems and continuous systems.Major changes in the new edition relate to Bose--Einstein condensation, the dynamics of the X-Y model and questions on phase transitions. Notes and remarks have been considerably augmented.

ADVAI\CES in communication, sensing, and computational power have Jed to an cxplosion of data. The size and varied formats for these datasets challenge existing techniqucs for transmission, storage, querying, display, and numerical manipula tion. This Ieads to the paradoxical situation where experiments or numerical com pulations produce rich, detailed inforrnation, for which, at this point, no adequate analysis tools exist. -Conference annow cement, Joint IDR-1/v A Workshop on Ideal Data Nepresentaticm, Minneapolis, R. De\'ore and A. Ron, cJ/gani ers Wavelct theory stands on the interface betwccn signal processing and harmonic analy sis, the rnathematical tools involved in digitizing continuous data with a vicw to storage, and thc synthesis proccss, recreating, for cxample, a picturc or time signal from stored data. The algorithms involved go under the name of tilter banks, and their spectacular efticiency derivcs in patt from the use of hidden self-similarity, relati\ c to somc scaling operation, in the daLJ. being analyzed. Observations or time signals are functions, and classes of functions make up linear spaces. Numcrical correlations add structure to thc spaccs at hand, Hilbcrt spaces. There are opcrators in the spaces deriving lrom the dis crcte data and others from the spaces of continuous signals. The first type arc good for computations, whilc the sccond retlect the real world. The operators between thc two are the focus of the prescnt monograph. Relations between operations in thc discrete xn Preface and continuous domains are studied as symbols."

For almost two decades this has been the classical textbook on applications of operator algebra theory to quantum statistical physics. It describes the general structure of equilibrium states, the KMS-condition and stability, quantum spin systems and continuous systems.
Major changes in the new edition relate to Bose--Einstein condensation, the dynamics of the X-Y model and questions on phase transitions. Notes and remarks have been considerably augmented.

The theme of the first Abel Symposium was operator algebras in a wide sense. In the last 40 years operator algebras have developed from a rather special discipline within functional analysis to become a central field in mathematics often described as "non-commutative geometry." It has branched out in several sub-disciplines and made contact with other subjects. The contributions to this volume give a state-of-the-art account of some of these sub-disciplines and the variety of topics reflect to some extent how the subject has developed. This is the first volume in a prestigious new book series linked to the Abel prize.

In this book we describe the elementary theory of operator algebras and parts of the advanced theory which are of relevance, or potentially of relevance, to mathematical physics. Subsequently we describe various applications to quantum statistical mechanics. At the outset of this project we intended to cover this material in one volume but in the course of develop ment it was realized that this would entail the omission ofvarious interesting topics or details. Consequently the book was split into two volumes, the first devoted to the general theory of operator algebras and the second to the applications. This splitting into theory and applications is conventional but somewhat arbitrary. In the last 15-20 years mathematical physicists have realized the importance of operator algebras and their states and automorphisms for problems of field theory and statistical mechanics. But the theory of 20 years aga was largely developed for the analysis of group representations and it was inadequate for many physical applications. Thus after a short honey moon period in which the new found tools of the extant theory were applied to the most amenable problems a longer and more interesting period ensued in which mathematical physicists were forced to redevelop the theory in relevant directions. New concepts were introduced, e. g. asymptotic abelian ness and KMS states, new techniques applied, e. g. the Choquet theory of barycentric decomposition for states, and new structural results obtained, e. g. the existence of a continuum of nonisomorphic type-three factors."