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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.

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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.- 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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