Tomita's Lectures on Observable Algebras in Hilbert Space

• Main Author: Inoue, Atsushi
• Format: eBook
• Language: English
• Published: Cham Springer Nature 2021; Springer International Publishing AG; Springer International Publishing
• Edition: 1
• Series: Lecture Notes in Mathematics
• Subjects: Algebra; Functional Analysis; Hilbert space; Mathematical Physics; Mathematics; Mathematics and Statistics; Mathematics. Analysis; Operator algebras; Operator Theory

This book is devoted to the study of Tomita's observable algebras, their structure and applications.It begins by building the foundations of the theory of T*-algebras and CT*-algebras, presenting the major results and investigating the relationship between the operator and vector representation...

Table of Contents:

• 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
• 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