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The Kadison-Singer Property

  • Book
  • © 2016

Overview

Authors:
  1. Marco Stevens

    1. Section of Analysis, Department of Mathematics, KU Leuven, Leuven, Belgium

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Part of the book series: SpringerBriefs in Mathematical Physics (BRIEFSMAPHY, volume 14)

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Table of contents (8 chapters)

  1. Front Matter

    Pages i-x
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  2. Introduction

    • Marco Stevens
    Pages 1-2

  3. Pure State Extensions in Linear Algebra

    • Marco Stevens
    Pages 3-10

  4. State Spaces and the Kadison-Singer Property

    • Marco Stevens
    Pages 11-21

  5. Maximal Abelian C\(^*\)-Subalgebras

    • Marco Stevens
    Pages 23-35

  6. Minimal Projections in Maximal Abelian von Neumann Algebras

    • Marco Stevens
    Pages 37-58

  7. Stone-Čech Compactification

    • Marco Stevens
    Pages 59-70

  8. The Continuous Subalgebra and the Kadison-Singer Conjecture

    • Marco Stevens
    Pages 71-84

  9. The Kadison-Singer Problem

    • Marco Stevens
    Pages 85-112

  10. Back Matter

    Pages 113-140
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Keywords

About this book

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.


Authors and Affiliations

  • Section of Analysis, Department of Mathematics, KU Leuven, Leuven, Belgium

    Marco Stevens

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