state space is non-empty
Proof : We first consider the case where is unital (http://planetmath.org/Ring), with identity element .
Let be the -subalgebra generated by and . Since is self-adjoint, is a comutative -algebra with identity element.
Regarding as an element of , attains a maximum at a point , since is compact. Hence, .
Also, and so is a norm one positive linear functional, i.e. is a state on .
Of course, is such that .
In case does not have an identity element we can consider its minimal unitization . By the preceding there is a state on satisfying the required . Now, we just need to take the restriction (http://planetmath.org/RestrictionOfAFunction) of to and this restriction is a state in satisfying the required .
|Title||state space is non-empty|
|Date of creation||2013-03-22 17:45:14|
|Last modified on||2013-03-22 17:45:14|
|Last modified by||asteroid (17536)|