A Spectral Theory Of Noncommuting Operators

Preface
Contents
1 Characteristic Polynomial in Several Variables
	1.1 Two Simple Examples
		1.1.1 The Lie Algebra su2
		1.1.2 The Infinite Dihedral Group
	1.2 General Properties
		1.2.1 The Coefficients of Characteristic Polynomial
		1.2.2 A Binomial Expansion in Three Variables
		1.2.3 Linear Factor
		1.2.4 Commuting Normal Matrices
		1.2.5 Unitary Invariants
	1.3 Determinantal Representation
		1.3.1 Irreducible Curves
		1.3.2 Dickson\'s Theorem
		1.3.3 Real-Zero Polynomials
2 Finite Dimensional Group Representations
	2.1 Basic Elements of Representation Theory
	2.2 Group Determinant
		2.2.1 An Old Theorem of Frobenius
		2.2.2 Group Determinant on a Generating Set
			2.2.2.1 The Alternating Group
			2.2.2.2 The Group GL3(Z/3Z)
	2.3 Abelian Groups
		2.3.1 Word Length Metric
		2.3.2 The Characteristic Polynomial Determines an Abelian Group
	2.4 The Coxeter Groups
		2.4.1 A Complete Invariant
		2.4.2 The Tits Representation
3 Finite Dimensional Lie Algebras
	3.1 Characteristic Polynomial for Lie Algebras
		3.1.1 Invariance Under the Automorphism Group
		3.1.2 The Irreducible Representations of sl2
		3.1.3 Semidirect Sum
	3.2 Simple Lie Algebras
		3.2.1 Root System
		3.2.2 Dynkin Diagram
		3.2.3 A Classification by Characteristic Polynomials
	3.3 Solvable Lie Algebras
		3.3.1 Spectral Matrix
		3.3.2 Spectral Invariants
4 Projective Spectrum in Banach Algebras
	4.1 Joint Spectra
		4.1.1 The Koszul Complex
		4.1.2 Taylor Spectrum
		4.1.3 Fredholm Tuples
		4.1.4 Essential Taylor Spectrum
		4.1.5 Harte Spectrum
	4.2 Projective Spectrum
		4.2.1 Connection with Taylor Spectrum
		4.2.2 Stein Domain
	4.3 Two Important C*-Algebras
		4.3.1 The Cuntz Algebra
		4.3.2 The Irrational Rotation Algebra
	4.4 Free Group von Neumann Algebras
		4.4.1 Noncommutative Probability Space
		4.4.2 Spectral Properties
5 The C*-Algebra of the Infinite Dihedral Group D∞
	5.1 Projective Spectrum of the Generators
		5.1.1 Regular Representation via the Bilateral Shift
		5.1.2 Connectedness of the Resolvent Set
	5.2 Two Projections in Generic Position
		5.2.1 Universal Projections in C*(D∞)
		5.2.2 Two Projection Matrices
	5.3 Fuglede–Kadison Determinant
		5.3.1 Basic Properties and Harpe–Skandalis Extension
		5.3.2 A Two-Variable Jacobi\'s Formula
		5.3.3 On the Fundamental Group of the Resolvent Set
		5.3.4 The FK Determinant of C*(D∞)
	5.4 An Application to Group of Intermediate Growth
		5.4.1 The Growth of Solvable and Nilpotent Groups
		5.4.2 The Grigorchuk Group
6 The Maurer–Cartan Form of Operator Pencils
	6.1 Curvature and Chern–Weil Theorem
		6.1.1 Connection and Curvature
		6.1.2 Invariant Linear Functional
		6.1.3 The Chern Class
		6.1.4 Chern–Simons Forms
		6.1.5 Chern–Simons Forms of Operator Pencils
	6.2 Trace Formula and Hyperplane Arrangement
		6.2.1 An Example with the Free Groups
		6.2.2 Theorems of Arnold and Brieskorn
		6.2.3 Abelian Banach Algebras
	6.3 Jacobi\'s Formula in Higher Orders
		6.3.1 su2 and the Chern Character
		6.3.2 An Extension to General Matrices
	6.4 A Note on Cyclic Cohomology
		6.4.1 Hochschild Cohomology and Cyclic Cocycle
		6.4.2 A Map into the de Rham Cohomology
		6.4.3 Cyclic Cocycles on the Irrational Rotation Algebra
7 Hermitian Metrics on the Resolvent Set
	7.1 Hermitian Vector Bundle
		7.1.1 The Dolbeault Operator
		7.1.2 Hermitian Metric
		7.1.3 Kähler Metric and the Ricci Form
	7.2 The Fundamental Form of Operator Pencils
		7.2.1 Operator-Valued Differential Forms
		7.2.2 Kähler Metric on the Resolvent Set
	7.3 The Issue of Completeness
		7.3.1 Distance in the Resolvent Set
		7.3.2 Another Example with D∞
	7.4 The Fundamental Form of a Single Operator
		7.4.1 Two Examples
		7.4.2 Ricci Curvature and Eigenvector
		7.4.3 Non-Euclidean Circles
	7.5 Extremal Equation and the Unilateral Shift
		7.5.1 Variational Calculus
		7.5.2 Inner Functions and the Extremal Length of Circles
	7.6 The Power Set of Quasinilpotent Operators
		7.6.1 Gauging the Singularities
		7.6.2 The Volterra Operators
		7.6.3 A Touch on the Hyper-Invariant Subspace
8 Compact Operators and Kernel Bundles
	8.1 The Fredholm Determinant and Logarithmic Integral
		8.1.1 The Argument Principle for Operator Functions
		8.1.2 The Trace of Residues
	8.2 The Projective Spectrum of Compact Operators
		8.2.1 Thin Set
		8.2.2 Normal Compact Operators
	8.3 Kernel Bundles
		8.3.1 On the Equivalence of Holomorphic Bundles
		8.3.2 Cowen–Douglas Operators
		8.3.3 The Kernel Bundle of Compact Operators
		8.3.4 An Example with Rank-1 Projections
		8.3.5 A Criterion for the Unitary Equivalence
9 Weak Containment and Amenability
	9.1 Locally Compact Groups
		9.1.1 Convolution
		9.1.2 Revisit the Regular Representation
	9.2 Weak Containment
		9.2.1 Some General Properties
		9.2.2 The Weak Containment of the Trivial Representation
	9.3 Amenability
		9.3.1 Invariant Mean
		9.3.2 The Markov–Kakutani Theorem
		9.3.3 A Spectral Description of Amenability
	9.4 Haagerup Groups and Kazhdan\'s Property (T)
		9.4.1 Definitions and Examples
		9.4.2 A Description in Several Variables
10 Self-similarity and Julia Sets
	10.1 Some Basics of Complex Dynamics
		10.1.1 Complex Dynamics in Cn
		10.1.2 The Green Function
		10.1.3 Complex Dynamics in Pn and the Indeterminacy Sets
	10.2 Self-similar Group Representations
		10.2.1 Reshuffling the Dyatic Intervals
		10.2.2 Three Illuminating Examples
	10.3 Renormalization Maps
		10.3.1 The Infinite Dihedral Group
		10.3.2 The Lamplighter Group
		10.3.3 The Grigorchuk Group
	10.4 The Julia Set of D∞
		10.4.1 Determining the Indeterminacy Set
		10.4.2 Projective Spectrum and the Julia Set
		10.4.3 The Limit of Iterations
References
Index