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4. 1 Bergman-Toeplitz Operators Over Bounded Domains 242 4. 2
Hardy-Toeplitz Operators Over Strictly Domains Pseudoconvex 250
Groupoid C* -Algebras 4. 3 256 4. 4 Hardy-Toeplitz Operators Over
Tubular Domains 267 4. 5 Bergman-Toeplitz Operators Over Tubular
Domains 278 4. 6 Hardy-Toeplitz Operators Over Polycircular Domains
284 4. 7 Bergman-Toeplitz Operators Over Polycircular Domains 290
4. 8 Hopf C* -Algebras 299 4. 9 Actions and Coactions on C*
-Algebras 310 4. 10 Hardy-Toeplitz Operators Over K-circular
Domains 316 4. 11 Hardy-Toeplitz Operators Over Symmetric Domains
325 4. 12 Bergman-Toeplitz Operators Over Symmetric Domains 361 5.
Index Theory for Multivariable Toeplitz Operators 5. 0 Introduction
371 5. 1 K-Theory for Topological Spaces 372 5. 2 Index Theory for
Strictly Pseudoconvex Domains 384 5. 3 C*-Algebras K-Theory for 394
5. 4 Index Theory for Symmetric Domains 400 5. 5 Index Theory for
Tubular Domains 432 5. 6 Index Theory for Polycircular Domains 455
References 462 Index of Symbols and Notations 471 In trod uction
Toeplitz operators on the classical Hardy space (on the I-torus)
and the closely related Wiener-Hopf operators (on the half-line)
form a central part of operator theory, with many applications e.
g. , to function theory on the unit disk and to the theory of
integral equations.
Aconferenceon"NoncommutativeGeometryandtheStandardModelof-
ementaryParticlePhysics"washeldattheHesselbergAcademy(innorthern
Bavaria, Germany) during the week of March 14-19, 1999. The aim of
the conference was to give a systematic exposition of the
mathematical foun- tions and physical applications of
noncommutative geometry, along the lines developedbyAlainConnes.
Theconferencewasactuallypartofacontinuing series of conferences at
the Hesselberg Academy held every three years and devoted to
important developments in mathematical ?elds, such as geom-
ricanalysis, operatoralgebras, indextheory,
andrelatedtopicstogetherwith their applications to mathematical
physics. The participants of the conference included mathematicians
from fu- tional analysis, di?erential geometry and operator
algebras, as well as - perts from mathematical physics interested
in A. Connes' approach towards the standard model and other
physical applications. Thus a large range of topics, from
mathematical foundations to recent physical applications, could
becoveredinasubstantialway. Theproceedingsofthisconference,
organized in a coherent and systematic way, are presented here. Its
three chapters c- respond to the main areas discussed during the
conference: Chapter1. Foundations of Noncommutative Geometry and
Basic Model Building Chapter2. The Lagrangian of the Standard Model
Derived from Nonc- mutative Geometry Chapter3. New Directions in
Noncommutative Geometry and Mathema- cal Physics During the
conference the close interaction between mathematicians and
mathematical physicists turned out to be quite fruitful and
enlightening for both sides. Similarly, it is hoped that the
proceedings presented here will be useful for mathematicians
interested in basic physical questions and for physicists aiming at
a more conceptual understanding of classical and qu- tum ?eld
theory from a novel mathematical point of view.
The outcome of a close collaboration between mathematicians and mathematical physicists, these Lecture Notes present the foundations of A. Connes noncommutative geometry, as well as its applications in particular to the field of theoretical particle physics. The coherent and systematic approach makes this book useful for experienced researchers and postgraduate students alike.
4. 1 Bergman-Toeplitz Operators Over Bounded Domains 242 4. 2
Hardy-Toeplitz Operators Over Strictly Domains Pseudoconvex 250
Groupoid C* -Algebras 4. 3 256 4. 4 Hardy-Toeplitz Operators Over
Tubular Domains 267 4. 5 Bergman-Toeplitz Operators Over Tubular
Domains 278 4. 6 Hardy-Toeplitz Operators Over Polycircular Domains
284 4. 7 Bergman-Toeplitz Operators Over Polycircular Domains 290
4. 8 Hopf C* -Algebras 299 4. 9 Actions and Coactions on C*
-Algebras 310 4. 10 Hardy-Toeplitz Operators Over K-circular
Domains 316 4. 11 Hardy-Toeplitz Operators Over Symmetric Domains
325 4. 12 Bergman-Toeplitz Operators Over Symmetric Domains 361 5.
Index Theory for Multivariable Toeplitz Operators 5. 0 Introduction
371 5. 1 K-Theory for Topological Spaces 372 5. 2 Index Theory for
Strictly Pseudoconvex Domains 384 5. 3 C*-Algebras K-Theory for 394
5. 4 Index Theory for Symmetric Domains 400 5. 5 Index Theory for
Tubular Domains 432 5. 6 Index Theory for Polycircular Domains 455
References 462 Index of Symbols and Notations 471 In trod uction
Toeplitz operators on the classical Hardy space (on the I-torus)
and the closely related Wiener-Hopf operators (on the half-line)
form a central part of operator theory, with many applications e.
g. , to function theory on the unit disk and to the theory of
integral equations.
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