rdf-syntax-grammar
mods:
  schema location: http://www.loc.gov/mods/v3 http://www.loc.gov/standards/mods/v3/mods-3-2.xsd
  title info:
    non sort: The
    title: Kadison-Singer Property
  name:
    name part: Stevens, Marco.
    role:
      role term: creator
    role:
      role term: author.
  name:
    name part: SpringerLink (Online service)
  type of resource: text
  origin info:
    place:
      place term: gw
    date issued: 2016
    issuance: monographic
  language:
    language term: eng
  physical description:
    form: electronic
    form: electronic resource
    reformatting quality: access
    form: remote
    extent: X, 140 p. online resource.
  abstract: 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.
  table of 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 -- Extensions of pure states -- Properties of extensions and restrictions -- Maximal abelian C*-subalgebras -- 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 -- 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.
  note: by Marco Stevens.
  subject:
    topic: Mathematics
  subject:
    topic: Functional analysis
  subject:
    topic: Operator theory
  subject:
    topic: Mathematical physics
  subject:
    topic: Physics
  subject:
    topic: Mathematics
  subject:
    topic: Mathematical Physics
  subject:
    topic: Operator Theory
  subject:
    topic: Mathematical Methods in Physics
  subject:
    topic: Functional Analysis
  classification: QA401-425
  classification: QC19.2-20.85
  classification: 530.15
  related item:
    title info:
      title: Springer eBooks
  related item:
    title info:
      title: SpringerBriefs in Mathematical Physics, 14
  identifier: 9783319477022
  identifier: http://dx.doi.org/10.1007/978-3-319-47702-2
  record info:
    record creation date: 161107
    record change date: 20161107055509.0
    record identifier: 978-3-319-47702-2