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.
