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Positive linear functional

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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

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f ( v ) ≥ 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 Cc(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
    • for all f in Cc(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 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+.

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

    Cauchy-Schwarz inequality

    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

    | ρ ( b ∗ a ) | 2 ≤ ρ ( a ∗ a ) ρ ( b ∗ b ) .

    References

    Positive linear functional Wikipedia
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