目錄
Introduction
1 Preliminaries
1.1 Operators in Hilbert space
1.1.1 Notation
1.1.2 Topologies of B(H)
1.1.3 Self-adjoint operators
1.1.4 Numerical range
1.1.5 The Bochner integral
1.2 Frechet derivativc
1.3 von Neumann algebras
1.3.1 Basic properties of von Neumann algebras
1.3.2 Projections in von Neumann algebras
1.3.3 Semifinite von Neumann algebras
1.3.4 Operators affiliated with a von Neumann algebra
1.3.5 Generalized s-numbers
1.3.6 Non-commutative L-spaces
1.3.7 Holomorphic functional calculus
1.3.8 Invariant operator ideals in semifinite von Neumann algebras
1.4 Integration of operator-valued functions
1.5 Theory of i-Fredhalm operators
1.5.1 Definition and elementary properties of T-Fredholm operators
1.5.2 The semifinite Fredholm alternative
1.5.3 The semifinite Atkinson theorem
1.5.4 Properties of T--Fredholm operators
1.5.5 Skew-corner T-Fredholrn operators
1,5.6 Essential codimension of two projections
1.5.7 The Carey-Phillips theorem
1.6 Spectral flow in semifinite von Neumann algebras
1.7 Fuglede-Kadison's determinant in sernifinite yon Neumann algebras
1.7.1 de la Harpe-Seandalis determinant
1.7.2 Technical lemmas
1.7.3 Definition of Fuglede-Kadison determinant and its properties
1.8 The Brown measure
1.8.1 Weyl functions
1.8.2 The Weierstrass function
1.8.3 Subharmonic functions
1.8.4 Technical results
1.8.5 The Brown measure
1.8.6 The Lidskii theorem for the Brown measure
1.8.7 Additional properties of the Brown measure
2 Spectrality of Dixmier trace
2.1 The Dixmier trace in sermifinite yon Neumann algebras
2.1.1 The Dixmier traces in semifinite von Neumann algebras
2.1.2 Measurability of operators
2.2 Lidskii formula for Dixmier traces
2.2.1 Spectral characterization of sums of commutators
2 2.2 The Lidskii formula for the Dixmier trace
3 Spectral shift function in yon Neumann algebras
3.1 Spectral shift function for trace class perturbations
3.1.1 Krein's trace formula: resolvent perturbations
3.1.2 The Krein trace formula: general case
3.2 Multiple operator integrals in yon Neumann algebras
3.2.1 BS representations
3.2.2 Multiple operator integrals
3.3 Higher order Frechet differenTiabiLity
3.4 Spectral shift and spectral averaging in semifinite yon Neumann algebras
4 Spectral shift function and spectral flow
4.1 Prelinfinary results
4.1.1 Self-adjoint operators with T-compact resolvent
4.1.2 Difference quotients and double operator integrals
4.1.3 Some continuity and differentinbfiity properties of operator functions138
4.1.4 A class Fa,b(N1, T) of T-Fredholm operators
4.2 The spectral shift function for operators with compact resolvent
4.2.1 The unbounded case
4.2.2 The bounded case
4.3 Spectral flow
4.3.1 The spectral flow function
4.3.2 Spectral flow one-forms: unbounded case
4.3.3 Spectral flow one-forms: bounded case
4.3.4 The first formula for spectral flow
4.3.5 Spectral flow in the unbounded case
4.3.6 The spectral flow formulae in the Z-summable spectral triple case
4.3.7 Recovering n-invariants
Concluding remarks
Bibliography
Index
編輯手記
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