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E-raamat: Singular Traces: Theory and Applications

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Singular Traces: Theory and ApplicationsSuurem pilt
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This book is the first complete study and monograph dedicated to singular traces. The text mathematically formalises the study of traces in a self contained theory of functional analysis. Extensive notes will treat the historical development. The final section will contain the most complete and concise treatment known of the integration half of Connes' quantum calculus. Singular traces are traces on ideals of compact operators that vanish on the subideal of finite rank operators. Singular traces feature in A. Connes' interpretation of noncommutative residues. Particularly the Dixmier trace, which generalises the restricted Adler-Manin-Wodzicki residue of pseudo-differential operators and plays the role of the residue for a new catalogue of 'geometric' spaces, including Connes-Chamseddine standard models, Yang-Mills action for quantum differential forms, fractals, isospectral deformations, foliations and noncommutative index theory. The theory of singular traces has been studied after Connes' application to non-commutative geometry and physics by various authors. Recent work by Nigel Kalton and the authors has advanced the theory of singular traces. Singular traces can be equated to symmetric functionals of symmetricsequence or function spaces, residues of zeta functions and heat kernel asymptotics, and characterised by Lidksii and Fredholm formulas. The traces and formulas used in noncommutative geometry are now completely understood in this theory, with surprising new mathematical and physical consequences. For mathematical readers the text offers fundamental functional analysis results and, due to Nigel Kalton's contribution, a now complete theory of traces on compact operators. For mathematical physicists and o

"Steven Lord, University of Adelaide, Australia; Fedor Sukochev and Dmitriy Zanin, University of New South Wales, Sydney, Australia. "
Preface v
Introduction 1(14)
I Preliminary Material
1 What is a Singular Trace?
15(23)
1.1 Compact Operators
15(7)
1.2 Calkin Correspondence
22(6)
1.3 Examples of Traces
28(6)
1.3.1 The Canonical Trace
29(1)
1.3.2 The Dixmier Trace
29(4)
1.3.3 Lidskii Formulation of Traces
33(1)
1.4 Notes
34(4)
2 Preliminaries on Symmetric Operator Spaces
38(41)
2.1 Von Neumann Algebras
38(4)
2.2 Semifinite Normal Traces
42(4)
2.3 Generalized Singular Value Function
46(7)
2.4 Calkin Correspondence in the Semifinite Setting
53(3)
2.5 Symmetric Operator Spaces
56(3)
2.6 Examples of Symmetric Operator Spaces
59(9)
2.7 Traces on Symmetric Operator Spaces
68(2)
2.8 Notes
70(9)
II General Theory
3 Symmetric Operator Spaces
79(28)
3.1 Introduction
79(1)
3.2 Submajorization in the Finite-dimensional Setting
80(3)
3.3 Hardy-Littlewood(-Polya) Submajorization
83(5)
3.4 Uniform Submajorization
88(9)
3.5 Symmetric Operator Spaces from Symmetric Function Spaces
97(4)
3.6 Symmetric Function Spaces from Symmetric Sequence Spaces
101(3)
3.7 Notes
104(3)
4 Symmetric Functionals
107(46)
4.1 Introduction
107(2)
4.2 Jordan Decomposition of Symmetric Functionals
109(5)
4.3 Lattice Structure on the Set of Symmetric Functionals
114(3)
4.4 Lifting of Symmetric Functionals
117(3)
4.5 Figiel-Kalton Theorem
120(3)
4.6 Existence of Symmetric Functionals
123(7)
4.7 Existence of Fully Symmetric Functionals
130(3)
4.8 The Sets of Symmetric and Fully Symmetric Functionals are Different
133(9)
4.9 Symmetric Functionals on Symmetric Operator Spaces
142(4)
4.10 How Large is the Set of Symmetric Functionals?
146(6)
4.11 Notes
152(1)
5 Commutator Subspace
153(41)
5.1 Introduction
153(2)
5.2 Normal Operators in the Commutator Subspace
155(7)
5.3 Normal Operators in the Closed Commutator Subspace
162(6)
5.4 Subharmonic Functions on Matrix Algebras
168(5)
5.5 Quasi-nilpotent Operators Belong to the Commutator Subspace
173(9)
5.6 Description of the Commutator Subspace
182(5)
5.7 Commutator Subspace of the Weak Ideal
187(5)
5.8 Notes
192(2)
6 Dixmier Traces
194(31)
6.1 Introduction
194(2)
6.2 Extended Limits
196(2)
6.3 Dixmier Traces on Lorentz Ideals
198(5)
6.4 Fully Symmetric Functionals on Lorentz Ideals are Dixmier Traces
203(3)
6.5 Dixmier Traces on Fully Symmetric Ideals of & pounds(H)
206(3)
6.6 Relatively Normal Functionals
209(5)
6.7 Wodzicki Representation of Dixmier Traces
214(3)
6.8 Notes
217(8)
III Traces on Lorentz Ideals
7 Lidskii Formulas for Dixmier Traces on Lorentz Ideals
225(19)
7.1 Introduction
225(1)
7.2 Distribution Formulas for Dixmier Traces
226(6)
7.3 Lidskii Formulas for Dixmier Traces
232(3)
7.4 Special Cases and Counterexamples
235(6)
7.5 Diagonal Formulas for Dixmier Traces Fail
241(1)
7.6 Notes
242(2)
8 Heat Kernel Formulas and function Residues
244(28)
8.1 Introduction
244(2)
8.2 Heat Kernel Functionals
246(6)
8.3 Fully Symmetric Functionals are Heat Kernel Functionals
252(4)
8.4 Generalized Heat Kernel Functionals
256(2)
8.5 Reduction of Generalized Heat Kernel Functionals
258(5)
8.6 Function Residues
263(5)
8.7 Not Every Dixmier Trace is a ζ-function Residue
268(3)
8.8 Notes
271(1)
9 Measurability in Lorentz Ideals
272(39)
9.1 Introduction
272(1)
9.2 Positive Dixmier Measurable Operators in Lorentz Ideals
273(3)
9.3 Positive Dixmier Measurable Operators in M1∞
276(5)
9.4 C-invariant Extended Limits
281(6)
9.5 Positive M-measurable Operators
287(4)
9.6 Additional Invariance of Dixmier Traces
291(6)
9.7 Measurable Operators in 1, ∞
297(3)
9.8 Notes
300(11)
IV Applications to Noncommutative Geometry
10 Preliminaries to the Applications
311(25)
10.1 Summary of Traces on £ ∞ and M1∞
311(6)
10.2 Pseudo-differential Operators and the Noncommutative Residue
317(13)
10.3 Pseudo-differential Operators on Manifolds
330(4)
10.4 Notes
334(2)
11 Trace Theorems
336(46)
11.1 Introduction
336(3)
11.2 Modulated Operators
339(6)
11.3 Laplacian Modulated Operators and Extension of the Noncommutative Residue
345(10)
11.4 Eigenvalues of Laplacian Modulated Operators
355(4)
11.5 Trace Theorem on Rd
359(3)
11.6 Trace Theorem on Closed Riemannian Manifolds
362(10)
11.7 Integration of Functions
372(8)
11.8 Notes
380(2)
12 Residues and Integrals in Noncommutative Geometry
382(38)
12.1 Introduction
382(3)
12.2 The Noncommutative Residue in Noncommutative Geometry
385(5)
12.3 The Integral in Noncommutative Geometry
390(6)
12.4 Example of Isospectral Deformations
396(9)
12.5 Example of the Noncommutative Torus
405(6)
12.6 Classical Limits
411(5)
12.7 Notes
416(4)
A Operator Results
420(9)
A.1 Matrix Results
420(2)
A.2 Operator Inequalities
422(7)
Bibliography 429(16)
Index 445
Steven Lord, University of Adelaide, Australia; Fedor Sukochev andDmitriy Zanin, University of New South Wales, Sydney, Australia.
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