eBook ISBN:  9780821876527 
Product Code:  CONM/62.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
eBook ISBN:  9780821876527 
Product Code:  CONM/62.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 

Book DetailsContemporary MathematicsVolume: 62; 1987; 544 ppMSC: Primary 46; Secondary 81; 82
This volume contains papers presented at the University of Iowa 1985 Summer Conference in honor of H.J. Borchers, N. M. Hugenholtz, R. V. Kadison, and D. Kastler and gives a systematic, uptodate treatment of the fruitful interaction that the last two decades have brought between operator algebras and mathematical physics. Special attention is paid to an overview of the algebraic approach to quantum field theory and in particular, to quantum statistical mechanics. More than half the papers culminate with a presentation of new results which have not appeared previously in journals, and with a few exceptions, these new results are presented with complete proofs.
This book is addressed to graduate students and researchers working in a broad spectrum of areas in mathematics and mathematical physics. Functional analysis, operator algebras, operator theory, differential geometry, cyclic cohomology, \(K\)theory, and index theory are applied to questions in the quantum theory of fields and statistical mechanics. The individual papers are selfcontained, but the reader should have some familiarity with the basic concepts of functional analysis and operator theory, although no physics background is assumed.

Table of Contents

I. Biographical Sketches [ MR 878371 ]

Klaus Fredenhagen — A biographical sketch of HansJürgen Borchers [ MR 878372 ]

Marinus Winnink — A biographical sketch of N. M. Hugenholtz [ MR 878373 ]

Erling Størmer — A biographical sketch of Richard V. Kadison [ MR 878374 ]

Rudolf Haag — A biographical sketch of Daniel Kastler [ MR 878375 ]

II. The Algebraic Approach to Quantum Theory of Fields and Statistical Mechanics [ MR 878371 ]

Huzihiro Araki — Bogoliubov automorphisms and Fock representations of canonical anticommutation relations [ MR 878376 ]

H.J. Borchers — On the interplay between spectrum condition and locality in quantum field theory [ MR 878377 ]

K. Fredenhagen — The structure of local algebras of observables [ MR 878378 ]

N. M. Hugenholtz — How the $C^\ast $algebraic formulation of statistical mechanics helps understanding the approach to equilibrium [ MR 878379 ]

Gérard G. Emch — KMS structures in geometric quantization [ MR 878380 ]

Shôichirô Sakai — Perturbations and KMS states in $C^\ast $dynamical systems (generalization of the absence theorem of phase transition to continuous quantum systems) [ MR 878381 ]

III. Methods from Differential Geometry and $K$theory [ MR 878371 ]

Daniel Kastler — ${\bf Z}/2$graded cyclic cohomology within the differential envelope [ MR 878382 ]

Alain Connes and Marc A. Rieffel — YangMills for noncommutative twotori [ MR 878383 ]

Marc A. Rieffel — Nonstable $K$theory and noncommutative tori [ MR 878384 ]

IV. Positivity in Operator Algebras [ MR 878371 ]

William Arveson — Nonlinear states on $C^\ast $algebras [ MR 878385 ]

Erling Størmer — Cones of positive maps [ MR 878386 ]

Jun Tomiyama — On the geometry of positive maps in matrix algebras [ MR 878387 ]

V. Dynamical Systems and Derivations [ MR 878371 ]

Derek W. Robinson — Differential operators on $C^\ast $algebras [ MR 878388 ]

Ola Bratteli — Derivations and free group actions [ MR 878389 ]

Ola Bratteli and Akitaka Kishimoto — Automatic continuity of derivations on eigenspaces [ MR 878390 ]

Shôichirô Sakai — Dynamics in continuous quantum systems [ MR 878391 ]

VI. Operators in Hilbert Spaces: Some Algebraic Methods [ MR 878371 ]

Joachim Cuntz and Nigel Higson — Kuiper’s theorem for Hilbert modules [ MR 878392 ]

Gerald W. Johnson and Michel L. Lapidus — Feynman’s operational calculus, generalized Dyson series and the Feynman integral [ MR 878393 ]

Robert T. Powers — An index theory for continuous semigroups of $\ast $endomorphisms of $\mathfrak {B}(\mathfrak {H})$ and ${\rm II}_1$ factors [ MR 878394 ]

Harald Upmeier — Multivariable Toeplitz operators, Jordan algebras and index theory [ MR 878395 ]

VII. $C^*$Algebras: Some Recent Results [ MR 878371 ]

Edward G. Effros — On multilinear completely bounded module maps [ MR 878396 ]

Berndt Brenken, Joachim Cuntz, George A. Elliott and Ryszard Nest — On the classification of noncommutative tori. III [ MR 878397 ]

VIII. Von Neumann Algebras [ MR 878371 ]

Roberto Longo — The joint modular structure of an inclusion of von Neumann algebras [ MR 878398 ]

Richard H. Herman and Vaughan F. R. Jones — Central sequences in crossed products [ MR 878399 ]


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This volume contains papers presented at the University of Iowa 1985 Summer Conference in honor of H.J. Borchers, N. M. Hugenholtz, R. V. Kadison, and D. Kastler and gives a systematic, uptodate treatment of the fruitful interaction that the last two decades have brought between operator algebras and mathematical physics. Special attention is paid to an overview of the algebraic approach to quantum field theory and in particular, to quantum statistical mechanics. More than half the papers culminate with a presentation of new results which have not appeared previously in journals, and with a few exceptions, these new results are presented with complete proofs.
This book is addressed to graduate students and researchers working in a broad spectrum of areas in mathematics and mathematical physics. Functional analysis, operator algebras, operator theory, differential geometry, cyclic cohomology, \(K\)theory, and index theory are applied to questions in the quantum theory of fields and statistical mechanics. The individual papers are selfcontained, but the reader should have some familiarity with the basic concepts of functional analysis and operator theory, although no physics background is assumed.

I. Biographical Sketches [ MR 878371 ]

Klaus Fredenhagen — A biographical sketch of HansJürgen Borchers [ MR 878372 ]

Marinus Winnink — A biographical sketch of N. M. Hugenholtz [ MR 878373 ]

Erling Størmer — A biographical sketch of Richard V. Kadison [ MR 878374 ]

Rudolf Haag — A biographical sketch of Daniel Kastler [ MR 878375 ]

II. The Algebraic Approach to Quantum Theory of Fields and Statistical Mechanics [ MR 878371 ]

Huzihiro Araki — Bogoliubov automorphisms and Fock representations of canonical anticommutation relations [ MR 878376 ]

H.J. Borchers — On the interplay between spectrum condition and locality in quantum field theory [ MR 878377 ]

K. Fredenhagen — The structure of local algebras of observables [ MR 878378 ]

N. M. Hugenholtz — How the $C^\ast $algebraic formulation of statistical mechanics helps understanding the approach to equilibrium [ MR 878379 ]

Gérard G. Emch — KMS structures in geometric quantization [ MR 878380 ]

Shôichirô Sakai — Perturbations and KMS states in $C^\ast $dynamical systems (generalization of the absence theorem of phase transition to continuous quantum systems) [ MR 878381 ]

III. Methods from Differential Geometry and $K$theory [ MR 878371 ]

Daniel Kastler — ${\bf Z}/2$graded cyclic cohomology within the differential envelope [ MR 878382 ]

Alain Connes and Marc A. Rieffel — YangMills for noncommutative twotori [ MR 878383 ]

Marc A. Rieffel — Nonstable $K$theory and noncommutative tori [ MR 878384 ]

IV. Positivity in Operator Algebras [ MR 878371 ]

William Arveson — Nonlinear states on $C^\ast $algebras [ MR 878385 ]

Erling Størmer — Cones of positive maps [ MR 878386 ]

Jun Tomiyama — On the geometry of positive maps in matrix algebras [ MR 878387 ]

V. Dynamical Systems and Derivations [ MR 878371 ]

Derek W. Robinson — Differential operators on $C^\ast $algebras [ MR 878388 ]

Ola Bratteli — Derivations and free group actions [ MR 878389 ]

Ola Bratteli and Akitaka Kishimoto — Automatic continuity of derivations on eigenspaces [ MR 878390 ]

Shôichirô Sakai — Dynamics in continuous quantum systems [ MR 878391 ]

VI. Operators in Hilbert Spaces: Some Algebraic Methods [ MR 878371 ]

Joachim Cuntz and Nigel Higson — Kuiper’s theorem for Hilbert modules [ MR 878392 ]

Gerald W. Johnson and Michel L. Lapidus — Feynman’s operational calculus, generalized Dyson series and the Feynman integral [ MR 878393 ]

Robert T. Powers — An index theory for continuous semigroups of $\ast $endomorphisms of $\mathfrak {B}(\mathfrak {H})$ and ${\rm II}_1$ factors [ MR 878394 ]

Harald Upmeier — Multivariable Toeplitz operators, Jordan algebras and index theory [ MR 878395 ]

VII. $C^*$Algebras: Some Recent Results [ MR 878371 ]

Edward G. Effros — On multilinear completely bounded module maps [ MR 878396 ]

Berndt Brenken, Joachim Cuntz, George A. Elliott and Ryszard Nest — On the classification of noncommutative tori. III [ MR 878397 ]

VIII. Von Neumann Algebras [ MR 878371 ]

Roberto Longo — The joint modular structure of an inclusion of von Neumann algebras [ MR 878398 ]

Richard H. Herman and Vaughan F. R. Jones — Central sequences in crossed products [ MR 878399 ]