A state Ψ on a C*-algebra A is a positive linear functionalMathworldPlanetmath Ψ:A, Ψ(a*a)0 for all aA, with unit norm. The norm of a positive linear functional is defined by

Ψ=supaA{|Ψ(a)|:a1}. (1)

For a unital C*-algebra, Ψ=Ψ(1I).

The space of states is a convex set. Let Ψ1 and Ψ2 be states, then the convex combination

λΨ1+(1-λ)Ψ2,λ[0,1], (2)

is also a state.

A state is pure if it is not a convex combination of two other states. Pure states are the extreme points of the convex set of states. A pure state on a commutative C*-algebra is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to a characterMathworldPlanetmath.

A state is called a tracial state if it is also a trace.

When a C*-algebra is represented on a Hilbert spaceMathworldPlanetmath , every unit vectorMathworldPlanetmath ψ determines a (not necessarily pure) state in the form of an expectation value,

Ψ(a)=ψ,aψ. (3)

In physics, it is common to refer to such states by their vector ψ rather than the linear functionalMathworldPlanetmathPlanetmath Ψ. The converseMathworldPlanetmath is not always true; not every state need be given by an expectation value. For example, delta functions (which are distributionsPlanetmathPlanetmathPlanetmath not functions) give pure states on C0(X), but they do not correspond to any vector in a Hilbert space (such a vector would not be square-integrable).


  • 1 G. Murphy, C*-Algebras and Operator Theory. Academic Press, 1990.
Title state
Canonical name State
Date of creation 2013-03-22 13:50:18
Last modified on 2013-03-22 13:50:18
Owner mhale (572)
Last modified by mhale (572)
Numerical id 8
Author mhale (572)
Entry type Definition
Classification msc 46L05
Related topic ExtensionAndRestrictionOfStates
Related topic AlgebraicQuantumFieldTheoriesAQFT
Defines pure state
Defines tracial state