Introduction to Operator Algebras

The following sections are included:

• Introduction

• Contents

The following sections are included:

• Banach spaces of operators on a Hilbert space

• Locally convex topologies in B(H)

• Von Neumann's double commutation theorem

• Tensor products of Von Neumann algebras

• Comparison of projections and central cover

• Kaplansky's density theorem

• Ideals in Von Neumann algebras

• Normal positive linear functionals

• Polar decomposition and orthogonal decomposition

• Radon-Nikodyn theorems

• The equivalence of the topologies s* and τ in a bounded ball

• Normal _* homomorphisms

• Comparison of cyclic projections and spatial _* isomorphic theorem

• σ-Finite Von Neumann algebras

The following sections are included:

• Definition and basic properties of C*-algebras

• Positive cones of C*-algebras

• States and the Gelfand-Naimark-Segal construction

• Approximate identities and quotient C*-algebras

• Extreme points of the unit ball and the existence of an identity

• Transitivity theorem and irreducible _* representations

• Pure states and regular maximal left ideals

• Ideals and quotient C*-algebras

• Hereditary C*-subalgebras

• Comparison, disjunction and quasi-equivalence of _* representations

• The enveloping Von Neumann algebra

• The multiplier algebra

• Finite dimensional C*-algebras

• The axioms for C*-algebras

• Real C*-algebras

The following sections are included:

• Tensor products of Banach spaces and cross-norms

• Tensor products of C*-algebras and the spatial C*-norm

• The maximal C*-norm

• States on algebraic tensor product

• The inequality λ(⋅) ≤ α₀(⋅) ≤ α(⋅) ≤ γ(⋅)

• Completely positive maps

• The inductive limit of C*–algebras

• Infinite tensor products of C*–algebras

• Nuclear C*–algebras

The following sections are included:

• Projections of norm one

• W*-algebras and their _* representations

• Tensor products of W*-algebras

• Completely additive functionals and singular functionals

• The characterizations of weakly compact subsets in predual

The following sections are included:

• Measure theory on locally compact Hausdorff spaces

• Stonean spaces

• Abelian W*-algebras

• _* Representations of abelian C*-algebras

The following sections are included:

• The classification of Von Neumann algebras

• An ergodic type theorem for Von Neumann algebras

• Finite Von Neumann algebras

• Properly infinite Von Neumann algebras

• Semi–finite Von Neumann algebras

• Purely infinite Von Neumann algebras

• Discrete (type (I )) Von Neumann algebras

• Continuous Von Neumann algebras and type (II) Von Neumann algebras

• The types of tensor products of Von Neumann algebras

The following sections are included:

• Dimension functions

• Hyperfinite type (II₁) factors

• Construction of factors of type (II) and type (III)

• The existences of non–hyperfinite type (II₁) factors and non–nuclear C*–algebras

The following sections are included:

• The KMS condition

• Tomita-Takesaki theory

• The modular automorphism group of a σ-finite W*-algebra

The following sections are included:

• Preliminaries

• The Arveson spectrum

• The Connes spectrum

• The Connes classification of type (III) factors (σ-finite case)

• Examples of type (III_λ) factors

The following sections are included:

• Polish spaces

• Borel subsets and Sousline subsets

• Borel maps and standard Borel spaces

• Borel cross sections

The following sections are included:

• The standard Borel structure of W(X*)

• Sequences of Borel choice functions

• The Borel spaces of Von Neumann algebras

• Borel subsets of factorial Borel space

The following sections are included:

• Measurable fields of Hilbert spaces

• Measurable fields of operators

• Measurable fields of Von Neumann algebras

• Decomposition of a Hilbert space into a direct integral

• The relations between a decomposable Von Neumann algebra and its components

• The constant fields of operators and Von Neumann algebras

• Borel subsets of the Borel space of Von Neumann algebras

• Borel subsets of the state space of a separable C*–algebra

The following sections are included:

• The spectrum of a C*–algebra

• Elementary C*–algebras and CCR ( liminary ) algebras

• GCR ( postliminary ) algebras and NGCR ( antiliminary ) algebras

• The existence of type (III) factorial _* representations of a NGCR algebra

• Type I C*–algebras

• Separable type I C*–algebras

The following sections are included:

• Choguet theory of boundary integrals on compact convex sets

• The C–measure and C–isomorphism of a state

• Extremal decomposition and central decomposition

• Ergodic decomposition and tracial decomposition

The following sections are included:

• The definition of (AF)-algebras

• Dimensions and isomorphic theorem

• The Bratteli diagrams of (AF)-algebras

• Ideals of (AF)-algebras

• Dimension groups

• Scaled dimension groups and stablly isomorphic theorem

• The tracial state space on an (AF)-algebra

The following sections are included:

• W*-crossed products

• Takesaki's duality theorem

• Group algebras and Group C*-algebras

• C*-crossed products

• Takai's duality theorem

• Some examples of crossed products

The following sections are included:

• The coupling constant

• Index for subfactors

• The fundamental construction

• The values of index for subfactors

The following sections are included:

• Appendix Weak Topology and Weak _* Topology

• References

• Natation Index

• Subject Index