The Kadison-Singer Property

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 e...

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Detalles Bibliográficos
Clasificación:Libro Electrónico
Autor principal: Stevens, Marco (Autor)
Autor Corporativo: SpringerLink (Online service)
Formato: Electrónico eBook
Idioma:Inglés
Publicado: Cham : Springer International Publishing : Imprint: Springer, 2016.
Edición:1st ed. 2016.
Colección:SpringerBriefs in Mathematical Physics, 14
Temas:
Mathematical physics.
Operator theory.
Functional analysis.
Mathematical Physics.
Operator Theory.
Mathematical Methods in Physics.
Functional Analysis.
Acceso en línea:Texto Completo
Tabla de Contenidos:
  • 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.

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