Editing Density matrix (section)

== C*-algebraic formulation of states ==
It is now generally accepted that the description of quantum mechanics in which all self-adjoint operators represent observables is untenable.<ref>See appendix,  {{Citation | last1=Mackey | first1=George Whitelaw | author1-link=George Mackey | title=Mathematical Foundations of Quantum Mechanics | publisher=[[Dover Publications]] | location=New York | series=Dover Books on Mathematics | isbn=978-0-486-43517-6 | year=1963}}</ref><ref>{{Citation | last1=Emch | first1=Gerard G. | title=Algebraic methods in statistical mechanics and quantum field theory | publisher=[[Wiley-Interscience]] | isbn=978-0-471-23900-0 | year=1972}}</ref> For this reason, observables are identified with elements of an abstract [[C*-algebra]] ''A'' (that is one without a distinguished representation as an algebra of operators) and [[state (functional analysis)|states]] are positive [[linear functional]]s on ''A''. However, by using the [[GNS construction]], we can recover Hilbert spaces that realize ''A'' as a subalgebra of operators.

Geometrically, a pure state on a C*-algebra ''A'' is a state that is an extreme point of the set of all states on ''A''. By properties of the GNS construction these states correspond to [[irreducible representation]]s of ''A''.

The states of the C*-algebra of [[compact operator]]s ''K''(''H'') correspond exactly to the density operators, and therefore the pure states of ''K''(''H'') are exactly the pure states in the sense of quantum mechanics.

The C*-algebraic formulation can be seen to include both classical and quantum systems. When the system is classical, the algebra of observables become an abelian C*-algebra. In that case the states become probability measures.