State (functional Analysis) - Jordan Decomposition

Jordan Decomposition

States can be viewed as noncommutative generalizations of probability measures. By Gelfand representation, every commutative C*-algebra A is of the form C₀(X) for some locally compact Hausdorff X. In this case, S(A) consists of positive Radon measures on X, and the pure states are the evaluation functionals on X.

More generally, the GNS construction shows that every state is, after a suitable representation, a vector state.

A bounded linear functional on a C*-algebra A is said to be self-adjoint if it is real-valued on the self-adjoint elements of A. Self-adjoint functionals are noncommutative analogues of signed measures.

The Jordan decomposition in measure theory says that every signed measure can be expressed as the difference of two positive measures supported on disjoint sets. This can be extended to the noncommutative setting.

Theorem Every self-adjoint f in A* can be written as f = f₊ − f₋ where f₊ and f₋ are positive functionals and ||f|| = ||f₊|| + ||f₋||.

A proof can be sketched as follows: Let Ω be the weak*-compact set of positive linear functionals on A with norm ≤ 1, and C(Ω) be the continuous functions on Ω. A can be viewed as a closed linear subspace of C(Ω) (this is Kadison's function representation). By Hahn–Banach, f extends to a g in C(Ω)* with ||g|| = ||f||.

Using results from measure theory quoted above, one has

where, by the self-adjointness of f, μ can be taken to be a signed measure. Write

a difference of positive measures. The restrictions of the functionals ∫ · dμ₊ and ∫ · dμ₋ to A has the required properties of f₊ and f₋. This proves the theorem.

It follows from the above decomposition that A* is the linear span of states.