Operator Algebras and Quantum Statistical Mechanics 1: C*- and W*-Algebras. Symmetry Groups. Decomposition of StatesIn 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. |
Contents
IV | 19 |
V | 25 |
VI | 32 |
VII | 39 |
VIII | 42 |
IX | 48 |
X | 54 |
XI | 58 |
XXXVI | 193 |
XXXVII | 202 |
XXXVIII | 209 |
XL | 233 |
XLI | 249 |
XLII | 264 |
XLIII | 269 |
XLIV | 290 |
XII | 61 |
XIII | 65 |
XIV | 71 |
XV | 75 |
XVI | 79 |
XVII | 83 |
XVIII | 84 |
XIX | 86 |
XX | 97 |
XXI | 102 |
XXII | 118 |
XXIII | 129 |
XXIV | 133 |
XXV | 136 |
XXVIII | 142 |
XXIX | 145 |
XXX | 152 |
XXXI | 157 |
XXXII | 159 |
XXXIII | 161 |
XXXIV | 163 |
XXXV | 184 |
Other editions - View all
Operator Algebras and Quantum Statistical Mechanics 1: C*- and W*-Algebras ... Ola Bratteli,Derek William Robinson No preview available - 2010 |
Common terms and phrases
A₁ A₂ abelian von Neumann analysis analytic elements assume Banach space barycenter Borel set bounded operators C*-algebra C*-algebra with identity characterization closed closure Co-semigroup commutant conditions are equivalent cone converges convex combination Corollary corresponding deduces defined Definition denote dense derivation dual ergodic establishes example exists F)-continuous factor finite following conditions G-abelian G-ergodic G-invariant group of automorphisms hence Hilbert space implies invariant invertible irreducible isometry isomorphism LC(H Lemma linear functional locally compact maximal measure modular Moreover morphism Neumann algebra nonzero norm normal o-weakly one-parameter group orthogonal measures polar decomposition projection Proposition prove quasi-local algebra result satisfies Section selfadjoint semigroup space H spectral spectrum strongly continuous subset subspace symmetry theory topology U₁ uniformly unique unitary elements unitary representation vector von Neumann algebra w₁ w₂ weak weakly
Popular passages
Page v - ... manuscript, and gave me much advice of which I have been glad to avail myself. I have also to acknowledge my indebtedness to Mr. WH Howell, Fellow of the Johns Hopkins University, who has corrected most of the proof-sheets, and prepared the index. H. NEWELL MA.KTIN.
Page 473 - Akemann, CA : The dual space of an operator algebra, Trans. Amer. Math. Soc.
Page 485 - Robinson, DW Statistical mechanics of quantum spin systems II, Commun. Math. Phys. 7 (1968), 337-348.
Page 482 - Landstad, MB Duality theory for covariant systems, Trans. Amer. Math. Soc. 248 ( 1979), 223-267.
Page 483 - MURRAY, FJ, and J. VON NEUMANN: On rings of operators. Ann. Math. 37, 116 — 229 (1936).