The Kadison-Singer Property

• Main Author: Stevens, Marco (Author)
• Corporate Author: SpringerLink (Online service)
• Format: Electronic eBook
• Language: English
• Published: Cham : Springer International Publishing : Imprint: Springer, 2016.
• Series: SpringerBriefs in Mathematical Physics, 14
• Subjects: Mathematics.; Functional analysis.; Operator theory.; Mathematical physics.; Physics.; Mathematical Physics.; Operator Theory.; Mathematical Methods in Physics.; Functional Analysis.

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...

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.