Positive linear functional

In mathematics, especially in functional analysis, a positive linear functional on an ordered vector space (V, ≤) is a linear functional f on V so that for all positive elements v of V, that is v≥0, it holds that

$f(v)\geq 0.$

In other words, a positive linear functional is guaranteed to take nonnegative values for positive elements. The significance of positive linear functionals lies in results such as Riesz–Markov–Kakutani representation theorem.

When V is a complex vector space, it is assumed that for all v≥0 , f(v) is real. As in the case when V is a C*-algebra with its partially ordered subspace of self-adjoint elements, sometimes a partial order is placed on only a subspace W of V, and the partial order does not extend to all of V, in which case the positive elements of V are the positive elements of W, by abuse of notation. This implies that for a C*-algebra, a positive linear functional sends any x in V equal to s*s for some s in V to a real number, which is equal to its complex conjugate, and therefore all positive linear functionals preserve the self-adjointness of such x. This property is exploited in the GNS construction to relate positive linear functionals on a C*-algebra to inner products.

Examples

• Consider, as an example of V, the C*-algebra of complex square matrices with the positive elements being the positive-definite matrices. The trace function defined on this C*-algebra is a positive functional, as the eigenvalues of any positive-definite matrix are positive, and so its trace is positive.

• Consider the Riesz space C_c(X) of all continuous complex-valued functions of compact support on a locally compact Hausdorff space X. Consider a Borel regular measure μ on X, and a functional ψ defined by

$\psi (f)=\int _{X}f(x)d\mu (x)\quad$

for all f in C_c(X). Then, this functional is positive (the integral of any positive function is a positive number). Moreover, any positive functional on this space has this form, as follows from the Riesz–Markov–Kakutani representation theorem.

Positive Linear Functionals (C*-algebras)

Let M be an operator system in a C*-algebra A with identity 1. Let M⁺ denote the set of positive elements in M.

A linear functional ρ on M is said to be positive if ρ(a) ≥ 0, for all a in M⁺.

Theorem. A linear functional ρ on M is positive if and only if ρ is bounded and ||ρ||=ρ(1).

States

If a positive linear functional ρ on M has norm 1, then ρ is described as a state. From the above theorem, ρ(1)=1.

The set of all states of M is convex, weak-* closed in the Banach dual space M^*. By the Banach-Alaoglu theorem, it is weak-* compact. Thus the set of all states of M with the weak-* topology forms a compact Hausdorff space, known as the state space of M.

Pure states

By the Krein-Milman theorem, the state space of M has extreme points. The extreme points of the state space are termed pure states.

Vector states

For H a Hilbert space and x in H, the equation ω_x(A) := <Ax,x> ( A in B(H) ), defines a positive linear functional on B(H). If ||x||=1, then ω_x is a state. If A is a C*-subalgebra of B(H) and M an operator system in A, then the restriction of ω_x to M defines a positive linear functional on M. The states of M that arise in this manner from unit vectors in H are termed vector states of M.

See also

• positive element