state space is non-empty

In this entry we prove the existence of states for every C*-algebra (

Theorem - Let 𝒜 be a C*-algebra. For every self-adjointPlanetmathPlanetmathPlanetmath ( element a𝒜 there exists a state ψ on 𝒜 such that |ψ(a)|=a.

Proof : We first consider the case where 𝒜 is unital (, with identity elementMathworldPlanetmath e.

Let be the C*-subalgebra generated by a and e. Since a is self-adjoint, is a comutative C*-algebra with identity element.

Thus, by the Gelfand-Naimark theoremMathworldPlanetmathPlanetmath, is isomorphicPlanetmathPlanetmathPlanetmath to C(X), the space of continuous functionsPlanetmathPlanetmath X for some compact set X.

Regarding a as an element of C(X), a attains a maximum at a point x0X, since X is compact. Hence, a=|a(x0)|.

The evaluation functionMathworldPlanetmath at x0,


is a multiplicative linear functional of C(X). Hence, evx0=1 and also |evx0(a)|=|a(x0)|=a.

We can now extend evx0 to a linear functionalMathworldPlanetmathPlanetmathPlanetmath ψ on 𝒜 such that ψ=evx0=1, using the Hahn-Banach theoremMathworldPlanetmath.

Also, ψ(e)=evx0(e)=1 and so ψ is a norm one positive linear functionalMathworldPlanetmath, i.e. ψ is a state on 𝒜.

Of course, ψ is such that |ψ(a)|=|evx0(a)|=a.

In case 𝒜 does not have an identity element we can consider its minimal unitization 𝒜~. By the preceding there is a state ψ~ on 𝒜~ satisfying the required . Now, we just need to take the restrictionPlanetmathPlanetmath ( of ψ~ to 𝒜 and this restriction is a state in 𝒜 satisfying the required .

Title state space is non-empty
Canonical name StateSpaceIsNonempty
Date of creation 2013-03-22 17:45:14
Last modified on 2013-03-22 17:45:14
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 14
Author asteroid (17536)
Entry type Theorem
Classification msc 46L30
Classification msc 46L05