Positive linear functional

In mathematics, more specifically 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

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.

Let M be a C*-algebra (more generally, 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⁺.

If ρ is a positive linear functional on a C*-algebra A, then one may define a semidefinite sesquilinear form on A by <a, b> := ρ(b^*a). Thus from the Cauchy–Schwarz inequality we have

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