Topics in Cyclic Theory (London Mathematical Society Student Texts)

Description

Noncommutative geometry combines themes from algebra, analysis and geometry and has significant applications to physics. This book focuses on cyclic theory, and is based upon the lecture courses by Daniel G. Quillen at the University of Oxford from 1988–92, which developed his own approach to the subject. The basic definitions, examples and exercises provided here allow non-specialists and students with a background in elementary functional analysis, commutative algebra and differential geometry to get to grips with the subject. Quillen's development of cyclic theory emphasizes analogies between commutative and noncommutative theories, in which he reinterpreted classical results of Hamiltonian mechanics, operator algebras and differential graded algebras into a new formalism. In this book, cyclic theory is developed from motivating examples and background towards general results. Themes covered are relevant to current research, including homomorphisms modulo powers of ideals, traces on noncommutative differential forms, quasi-free algebras and Chern characters on connections. Contents Introduction 1 Background Results 1.1 Graded Algebras 1.2 Derivations 1.3 Commutators and Traces 1.4 Tensor Algebras 1.5 Real Clifford Algebras 1.6 Lie Bracket 1.7 The Poisson Bracket 1.8 Extensions of Algebras via Modules 1.9 Deformations of the Standard Product 2 Cyclic Cocycles and Basic Operators 2.1 The Chain Complex 2.2 The λ and b Operators 2.3 Cyclic Cocycles on a Manifold 2.4 Double Complexes 2.5 The b′ and N Operators 2.6 Hochschild Cohomology 2.7 Vector Traces 2.8 Bianchi’s Identity 2.9 Projective Modules 2.10 Singular Homology 3 Algebras of Operators 3.1 The Gelfand Transform 3.2 Ideals of Compact Operators on Hilbert Space 3.3 Algebras of Operators on Hilbert Space 3.4 Fredholm Operators 3.5 Index Theory on the Circle via Toeplitz Operators 3.6 The Index Formula for Toeplitz Operators 3.7 Wallach’s Formula 3.8 Extensions of Commutative C*-Algebras 3.9 Idempotents and Generalized Toeplitz Operators 4 GNS Algebra 4.1 Idempotents and Dilations 4.2 GNS Theorem for States on a C*-Algebra 4.3 GNS Algebra 4.4 Stinespring’s Theorem 4.5 The Generalized Stinespring Theorem 4.6 Uniqueness of GNS(ρ) 4.7 Projective Hilbert Modules 4.8 Algebras Associated with the Continuous Functions on the Circle 4.9 Algebras Described by Universal Mapping Properties 4.10 The Universal GNS Algebra of the Tensor Algebra 4.11 The Cuntz Algebra 4.12 Fredholm Modules 5 Geometrical Examples 5.1 Fredholm Modules over the Circle 5.2 Heat Kernels on Riemannian Manifolds 5.3 Green’s Function 5.4 Maxwell’s Equation 5.5 Dirac Operators 5.6 Theta Summable Fredholm Modules 5.7 Duhamel’s Formula 5.8 Quantum Harmonic Oscillator 5.9 Chern Polynomials and Generating Functions 6 The Algebra of Noncommutative Differential Forms 6.1 Kahler Differentials on an Algebraic Curve 6.2 Homology of Kahler Differential Forms 6.3 Noncommutative Differential Forms ΩA 6.4 Ω1A as an A-bimodule 6.5 The Cuntz Algebra with Fedosov’s Product 6.6 Cyclic Cochains on the Cuntz Algebra 6.7 Tensor Algebra with the Fedosov Product 6.8 Completions 7 Hodge Decomposition and the Karoubi Operator 7.1 Hodge Decomposition on a Compact Riemann Surface 7.2 The b Operator and Hochschild Homology 7.3 The Karoubi Operator 7.4 Connes’s B-operator 7.5 The Hodge Decomposition 7.6 Harmonic Forms 7.7 Mixed Complexes in the Homology Setting 7.8 Homology of the Reduced Differential Forms 7.9 Cyclic Cohomology 7.10 Traces on RA and Cyclic Cocycles on A 8 Connections 8.1 Connections and Curvature on Manifolds 8.2 The Chern Character 8.3 Deforming Flat Connections 8.4 Universal Differentials 8.5 Connections on Modules over an Algebra 8.6 Derivations and Automorphisms 8.7 Lifting and Automorphisms of QA 9 Cocycles for a Commutative Algebra over a Manifold 9.1 Poisson Structures on a Manifold 9.2 Weyl Algebras 9.3 Representations of the Heisenberg Group 9.4 Quantum Trace Formula 9.5 The Poisson Bracket and Symbols 9.6 Cocycles Generated by Commutator Products 10 Cyclic Cochains 10.1 Traces Modulo Powers of an Ideal 10.2 Coalgebra 10.3 Quotienting by the Commutator Subspace 10.4 Bar Construction 10.5 Cochains with Values in an Algebra 10.6 Analogue of Ω1R for the Bar Construction 10.7 Traces Modulo an Ideal 10.8 Analogue of the Quotient by Commutators 10.9 Connes’s Chain and Cochain Bicomplexes 11 Cyclic Cohomology 11.1 Connes’s Double Cochain Complex 11.2 Connes’s S Operator 11.3 Connes’s Long Exact Sequence 11.4 A Homotopy Formula for Cocycles Associated with Traces 11.5 Universal Graded and Ungraded Cocycles 11.6 Deformations of Fredholm Modules 11.7 Homotopy Formulas 11.8 Cyclic Cocycles over the Circle 11.9 Connections over a Compact Manifold 11.10 The Trivial Bundle 11.11 Cocycles Arising from the Connection 11.12 Super Connections and Twisted Dirac Operators 12 Periodic Cyclic Homology 12.1 The X Complex and Periodic Cyclic Homology 12.2 X(A) for Commutative Differential Graded Algebras 12.3 The Canonical Filtration 12.4 The Hodge Approximation to Cyclic Theory References List of Symbols Index of Subjects