The Kadison-Singer Property
Springer International Publishing (Verlag)
978-3-319-47701-5 (ISBN)
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.
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-Cech compactification.- Stone-Cech 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.- TheAnderson 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.
Erscheinungsdatum | 02.12.2016 |
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Reihe/Serie | SpringerBriefs in Mathematical Physics |
Zusatzinfo | X, 140 p. |
Verlagsort | Cham |
Sprache | englisch |
Maße | 155 x 235 mm |
Themenwelt | Mathematik / Informatik ► Mathematik ► Analysis |
Naturwissenschaften ► Physik / Astronomie ► Theoretische Physik | |
Schlagworte | Anderson operator • C -algebras • C*-algebras • convergence of ultrafilters • Functional Analysis • Functional analysis and transforms • GNS-representation • Haar states • Hausdorff Spaces • Mathematical methods in physics • Mathematical Physics • Mathematics • mathematics and statistics • maximal abelian subalgebras • operator theory • Paving theorems • Stone-Cech Compactification • Stone-Čech compactification • Transitivity theorem • Tychonoff spaces • ultrafilters • von neumann algebras • Weaver's theorem • Weaver’s theorem |
ISBN-10 | 3-319-47701-3 / 3319477013 |
ISBN-13 | 978-3-319-47701-5 / 9783319477015 |
Zustand | Neuware |
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