The Kadison-Singer property / by Marco Stevens.

Format:

Book

Author/Creator:

Stevens, Marco., Author.

Series:

SpringerBriefs in Mathematical Physics, 2197-1757 ; 14

Language:

English

Subjects (All):

Mathematical physics.

Operator theory.

Physics.

Functional analysis.

Mathematical Physics.

Operator Theory.

Mathematical Methods in Physics.

Functional Analysis.

Local Subjects:

Mathematical Physics.

Operator Theory.

Mathematical Methods in Physics.

Functional Analysis.

Physical Description:

1 online resource (X, 140 p.)

Edition:

1st ed. 2016.

Place of Publication:

Cham : Springer International Publishing : Imprint: Springer, 2016.

Summary:

This book gives a complete classification of all algebras with the Kadison-Singer property, when restricting to separable Hilbert spaces. The Kadison-Singer property deals with the following question: given a Hilbert space H and an abelian unital C*-subalgebra A of B(H), does every pure state on A extend uniquely to a pure state on B(H)? This question has deep connections to fundamental aspects of quantum physics, as is explained in the foreword by Klaas Landsman. The book starts with an accessible introduction to the concept of states and continues with a detailed proof of the classification of maximal Abelian von Neumann algebras, a very explicit construction of the Stone-Cech compactification and an account of the recent proof of the Kadison-Singer problem. At the end accessible appendices provide the necessary background material. This elementary account of the Kadison-Singer conjecture is very well-suited for graduate students interested in operator algebras and states, researchers who are non-specialists of the field, and/or interested in fundamental quantum physics.

Contents:

Introduction.-Pure state extensions in linear algebra

Density operators and pure states

Extensions of pure states

State spaces and the Kadison-Singer property

States on C*-algebras

Pure states and characters

Properties of extensions and restrictions

Maximal abelian C*-subalgebras

Examples of maximal abelian C*-subalgebras

Minimal projections in maximal abelian von Neumann algebras

Unitary equivalence

Minimal projections

Subalgebras without minimal projections

Subalgebras with minimal projections

Classification

Stone-Čech compactification

Ultrafilters

Zero-sets

Ultra-topology.-Convergence of ultrafilters for Tychonoff spaces

Pushforward

Convergence of ultrafilters for compact Hausdorff spaces

Universal property

The continuous subalgebra and the Kadison-Singer conjecture

Total sets of states

Haar states

Projections in the continuous subalgebra

The Anderson operator

The Kadison-Singer conjecture

The Kadison-Singer problem

Real stable polynomials

Realizations of random matrices

Orthants and absence of zeroes

Weaver’s theorem

Paving theorems

Proof of the Kadison-Singer conjecture

Preliminaries

Linear algebra

Order theory

Topology

Complex analysis

Functional Analysis and Operator Algebras

Basic functional analysis

Hilbert spaces

C*-algebras

Von Neumann algebras

Additional material

Transitivity theorem

G-sets, M-sets and L-sets

GNS-representation

Miscellaneous

Notes and remarks

References.

Notes:

Includes bibliographical references at the end of each chapters.

ISBN:

3-319-47702-1