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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