Tomita's Lectures on Observable Algebras in Hilbert Space

Name: Tomita's Lectures on Observable Algebras in Hilbert Space
Brand: Springer
Price: 58.84 EUR
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Atsushi Inoue(Autor*in)

Springer (Verlag)

Erschienen am 1. März 2021

X, 190 Seiten

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978-3-030-68893-6 (ISBN)

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Intro
Preface
Contents
1 Introduction
2 Fundamentals of Observable Algebras
2.1 Q-algebras and T-algebras
2.2 Structure of CQ-algebras
2.3 Functional Calculus for Self-adjoint Trio Observables
2.4 -Automorphisms of Observable Algebras
2.5 Locally Convex Topologies on T-algebras
2.6 Commutants and Bicommutants of T-algebras
3 Density Theorems
3.1 Von Neumann Type Density Theorem
3.2 Kaplansky Type Density Theorem
4 Structure of CT-Algebras
4.1 Decomposition of CT-Algebras
4.2 Classification of CT-Algebras
4.3 Commutative Semisimple CT-Algebras
4.4 Some Results Obtained from T-Algebras
4.4.1 Projections Defined by T-Algebras
4.4.2 The Vector Representation of the CT-Algebra Generated by a T-Algebra
4.4.3 Construction of Semisimple CT-Algebras from a T-Algebra
4.4.4 Semisimplicity and Singularity of T-Algebras
4.4.5 A Natural Weight on the von Neumann Algebra Defined from a T-Algebra
4.4.6 Density of a -Subalgebra of a T-Algebra
5 Applications
5.1 Standard T-Algebras and the Tomita-Takesaki Theory
5.2 Admissible Invariant Positive Invariant Sesquilinear Forms on a -Algebra
5.2.1 T-Algebras Generated by Admissible i.p.s. Forms
5.2.2 Admissibility and Representability of i.p.s. Forms
5.2.3 Strongly Regular i.p.s. Forms
5.2.4 Decomposition of i.p.s. Forms
5.2.5 Regularity and Singularity of i.p.s. Forms
5.3 Positive Definite Generalized Functions in Lie Groups
5.4 Weights in C-Algebras
A Functional Calculus, Polar Decomposition and Spectral Resolution for Bounded Operators on a Hilbert Space
A.1 Spectrum
A.2 Continuous Functional Calculus of a Self-Adjoint Bounded Linear Operator
A.3 Polar Decomposition for a Bounded Linear Operator
A.4 Spectral Resolution for a Bounded Self-Adjoint Operator
A.5 Functional Calculus with Borel Functions
B Spectral Resolution of an Unbounded Self-Adjoint Operator and Polor Decomposition of a Closed Operator in a Hilbert Space
B.1 Basic Definitions and Results for Unbounded Linear Operators
B.2 Spectral Resolution of Unbounded Self-Adjoint Operators
B.3 Functional Calculus for Unbounded Self-Adjoint Operators
B.4 Polar Decomposition of Closed Linear Operators
C Banach -Algebras, C-Algebras, von Neumann Algebras and Locally Convex -Algebras
C.1 Banach -Algebras
C.2 C-Algebras and von Neumann Algebras
C.3 Density Theorems
C.4 Locally Convex -Algebras
References
Index

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