دانلود کتاب Singular Traces: Theory and Applications

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Preface\nIntroduction\nI Preliminary Material\n 1 What is a Singular Trace?\n 1.1 Compact Operators\n 1.2 Calkin Correspondence\n 1.3 Examples of Traces\n 1.3.1 The Canonical Trace\n 1.3.2 The Dixmier Trace\n 1.3.3 Lidskii Formulation of Traces\n 1.4 Notes\n 2 Preliminaries on Symmetric Operator Spaces\n 2.1 Von Neumann Algebras\n 2.2 Semifinite Normal Traces\n 2.3 Generalized Singular Value Function\n 2.4 Calkin Correspondence in the Semifinite Setting\n 2.5 Symmetric Operator Spaces\n 2.6 Examples of Symmetric Operator Spaces\n 2.7 Traces on Symmetric Operator Spaces\n 2.8 Notes\nII General Theory\n 3 Symmetric Operator Spaces\n 3.1 Introduction\n 3.2 Submajorization in the Finite-dimensional Setting\n 3.3 Hardy-Littlewood(-Polya) Submajorization\n 3.4 Uniform Submajorization\n 3.5 Symmetric Operator Spaces from Symmetric Function Spaces\n 3.6 Symmetric Function Spaces from Symmetric Sequence Spaces\n 3.7 Notes\n 4 Symmetric Functionals\n 4.1 Introduction\n 4.2 Jordan Decomposition of Symmetric Functionals\n 4.3 Lattice Structure on the Set of Symmetric Functionals\n 4.4 Lifting of Symmetric Functionals\n 4.5 Figiel-Kalton Theorem\n 4.6 Existence of Symmetric Functionals\n 4.7 Existence of Fully Symmetric Functionals\n 4.8 The Sets of Symmetric and Fully Symmetric Functionals are Different\n 4.9 Symmetric Functionals on Symmetric Operator Spaces\n 4.10 How Large is the Set of Symmetric Functionals?\n 4.11 Notes\n 5 Commutator Subspace\n 5.1 Introduction\n 5.2 Normal Operators in the Commutator Subspace\n 5.3 Normal Operators in the Closed Commutator Subspace\n 5.4 Subharmonic Functions on Matrix Algebras\n 5.5 Quasi-nilpotent Operators Belong to the Commutator Subspace\n 5.6 Description of the Commutator Subspace\n 5.7 Commutator Subspace of the Weak Ideal\n 5.8 Notes\n 6 Dixmier Traces\n 6.1 Introduction\n 6.2 Extended Limits\n 6.3 Dixmier Traces on Lorentz Ideals\n 6.4 Fully Symmetric Functionals on Lorentz Ideals are Dixmier Traces\n 6.5 Dixmier Traces on Fully Symmetric Ideals of L(H)\n 6.6 Relatively Normal Functionals\n 6.7 Wodzicki Representation of Dixmier Traces\n 6.8 Notes\nIII Traces on Lorentz Ideals\n 7 Lidskii Formulas for Dixmier Traces on Lorentz Ideals\n 7.1 Introduction\n 7.2 Distribution Formulas for Dixmier Traces\n 7.3 Lidskii Formulas for Dixmier Traces\n 7.4 Special Cases and Counterexamples\n 7.5 Diagonal Formulas for Dixmier Traces Fail\n 7.6 Notes\n 8 Heat Kernel Formulas and ζ-function Residues\n 8.1 Introduction\n 8.2 Heat Kernel Functionals\n 8.3 Fully Symmetric Functionals are Heat Kernel Functionals\n 8.4 Generalized Heat Kernel Functionals\n 8.5 Reduction of Generalized Heat Kernel Functionals\n 8.6 ζ-function Residues\n 8.7 Not Every Dixmier Trace is a ζ-function Residue\n 8.8 Notes\n 9 Measurability in Lorentz Ideals\n 9.1 Introduction\n 9.2 Positive Dixmier Measurable Operators in Lorentz Ideals\n 9.3 Positive Dixmier Measurable Operators in M1,8\n 9.4 C-invariant Extended Limits\n 9.5 Positive M-measurable Operators\n 9.6 Additional Invariance of Dixmier Traces\n 9.7 Measurable Operators in\n 9.8 Notes\nIV Applications to Noncommutative Geometry\n 10 Preliminaries to the Applications\n 10.1 Summary of Traces on L1,∞ and M1,∞\n 10.2 Pseudo-differential Operators and the Noncommutative Residue\n 10.3 Pseudo-differential Operators on Manifolds\n 10.4 Notes\n 11 Trace Theorems\n 11.1 Introduction\n 11.2 Modulated Operators\n 11.3 Laplacian Modulated Operators and Extension of the Noncommutative Residue\n 11.4 Eigenvalues of Laplacian Modulated Operators\n 11.5 Trace Theorem on Rd\n 11.6 Trace Theorem on Closed Riemannian Manifolds\n 11.7 Integration of Functions\n 11.8 Notes\n 12 Residues and Integrals in Noncommutative Geometry\n 12.1 Introduction\n 12.2 The Noncommutative Residue in Noncommutative Geometry\n 12.3 The Integral in Noncommutative Geometry\n 12.4 Example of Isospectral Deformations\n 12.5 Example of the Noncommutative Torus\n 12.6 Classical Limits\n 12.7 Notes\n A Operator Results\n A.1 Matrix Results\n A.2 Operator Inequalities\nBibliography\nIndex