Positive linear functional

Sufficient conditions for continuity of all positive linear functionals

There is a comparatively large class of ordered topological vector spaces on which every positive linear form is necessarily continuous. This includes all topological vector lattices that are sequentially complete.

Theorem Let X be an Ordered topological vector space with positive cone C \subseteq X and let \mathcal{B} \subseteq \mathcal{P}(X) denote the family of all bounded subsets of X. Then each of the following conditions is sufficient to guarantee that every positive linear functional on X is continuous:

1. C has non-empty topological interior (in X).
2. X is complete and metrizable and X = C - C.
3. X is bornological and C is a semi-complete strict \mathcal{B}-cone in X.
4. X is the inductive limit of a family \left(X_{\alpha} \right){\alpha \in A} of ordered Fréchet spaces with respect to a family of positive linear maps where X{\alpha} = C_{\alpha} - C_{\alpha} for all \alpha \in A, where C_{\alpha} is the positive cone of X_{\alpha}.

Continuous positive extensions

The following theorem is due to H. Bauer and independently, to Namioka.

:Theorem: Let X be an ordered topological vector space (TVS) with positive cone C, let M be a vector subspace of E, and let f be a linear form on M. Then f has an extension to a continuous positive linear form on X if and only if there exists some convex neighborhood U of 0 in X such that \operatorname{Re} f is bounded above on M \cap (U - C).

:Corollary: Let X be an ordered topological vector space with positive cone C, let M be a vector subspace of E. If C \cap M contains an interior point of C then every continuous positive linear form on M has an extension to a continuous positive linear form on X.

:Corollary: Let X be an ordered vector space with positive cone C, let M be a vector subspace of E, and let f be a linear form on M. Then f has an extension to a positive linear form on X if and only if there exists some convex absorbing subset W in X containing the origin of X such that \operatorname{Re} f is bounded above on M \cap (W - C).

Proof: It suffices to endow X with the finest locally convex topology making W into a neighborhood of 0 \in X.

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 \mathrm{C}{\mathrm{c}}(X) of all continuous complex-valued functions of compact support on a locally compact Hausdorff space X. Consider a Borel regular measure \mu on X, and a functional \psi defined by \psi(f) = \int_X f(x) d \mu(x) \quad \text{ for all } f \in \mathrm{C}{\mathrm{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 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 \rho on M is said to be positive if \rho(a) \geq 0, for all a \in M^+. :Theorem. A linear functional \rho on M is positive if and only if \rho is bounded and |\rho| = \rho(1).

Cauchy–Schwarz inequality

If \rho is a positive linear functional on a C*-algebra A, then one may define a semidefinite sesquilinear form on A by \langle a,b\rangle = \rho(b^{\ast}a). Thus from the Cauchy–Schwarz inequality we have \left|\rho(b^{\ast}a)\right|^2 \leq \rho(a^{\ast}a) \cdot \rho(b^{\ast}b).

Applications to economics

Given a space C, a price system can be viewed as a continuous, positive, linear functional on C.

References

Bibliography

• Kadison, Richard, Fundamentals of the Theory of Operator Algebras, Vol. I : Elementary Theory, American Mathematical Society. .

References

1. Murphy, Gerard. "C*-Algebras and Operator Theory". Academic Press, Inc..