Positive linear functional

In mathematics, more specifically in functional analysis, a positive linear functional on an ordered vector space $$(V, \leq)$$ is a linear functional $$f$$ on $$V$$ so that for all positive elements $$v \in V,$$ that is $$v \geq 0,$$ 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 $$v\ge0,$$ $$f(v)$$ 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 $$W\subseteq V,$$ 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 $$s^{\ast}s$$ 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.

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 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 $$C$$.