# Positive linear functional

In mathematics, more specifically in functional analysis, a positive linear functional on an ordered vector space $\displaystyle{ (V, \leq) }$ is a linear functional $\displaystyle{ f }$ on $\displaystyle{ V }$ so that for all positive elements $\displaystyle{ v \in V, }$ that is $\displaystyle{ v \geq 0, }$ it holds that $\displaystyle{ f(v) \geq 0. }$

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 $\displaystyle{ V }$ is a complex vector space, it is assumed that for all $\displaystyle{ v\ge0, }$ $\displaystyle{ f(v) }$ is real. As in the case when $\displaystyle{ V }$ is a C*-algebra with its partially ordered subspace of self-adjoint elements, sometimes a partial order is placed on only a subspace $\displaystyle{ W\subseteq V, }$ and the partial order does not extend to all of $\displaystyle{ V, }$ in which case the positive elements of $\displaystyle{ V }$ are the positive elements of $\displaystyle{ W, }$ by abuse of notation. This implies that for a C*-algebra, a positive linear functional sends any $\displaystyle{ x \in V }$ equal to $\displaystyle{ s^{\ast}s }$ for some $\displaystyle{ s \in V }$ to a real number, which is equal to its complex conjugate, and therefore all positive linear functionals preserve the self-adjointness of such $\displaystyle{ x. }$ 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.[1] This includes all topological vector lattices that are sequentially complete.[1]

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

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

## Continuous positive extensions

The following theorem is due to H. Bauer and independently, to Namioka.[1]

Theorem:[1] Let $\displaystyle{ X }$ be an ordered topological vector space (TVS) with positive cone $\displaystyle{ C, }$ let $\displaystyle{ M }$ be a vector subspace of $\displaystyle{ E, }$ and let $\displaystyle{ f }$ be a linear form on $\displaystyle{ M. }$ Then $\displaystyle{ f }$ has an extension to a continuous positive linear form on $\displaystyle{ X }$ if and only if there exists some convex neighborhood $\displaystyle{ U }$ of $\displaystyle{ 0 }$ in $\displaystyle{ X }$ such that $\displaystyle{ \operatorname{Re} f }$ is bounded above on $\displaystyle{ M \cap (U - C). }$
Corollary:[1] Let $\displaystyle{ X }$ be an ordered topological vector space with positive cone $\displaystyle{ C, }$ let $\displaystyle{ M }$ be a vector subspace of $\displaystyle{ E. }$ If $\displaystyle{ C \cap M }$ contains an interior point of $\displaystyle{ C }$ then every continuous positive linear form on $\displaystyle{ M }$ has an extension to a continuous positive linear form on $\displaystyle{ X. }$
Corollary:[1] Let $\displaystyle{ X }$ be an ordered vector space with positive cone $\displaystyle{ C, }$ let $\displaystyle{ M }$ be a vector subspace of $\displaystyle{ E, }$ and let $\displaystyle{ f }$ be a linear form on $\displaystyle{ M. }$ Then $\displaystyle{ f }$ has an extension to a positive linear form on $\displaystyle{ X }$ if and only if there exists some convex absorbing subset $\displaystyle{ W }$ in $\displaystyle{ X }$ containing the origin of $\displaystyle{ X }$ such that $\displaystyle{ \operatorname{Re} f }$ is bounded above on $\displaystyle{ M \cap (W - C). }$

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

## Examples

Consider, as an example of $\displaystyle{ V, }$ 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 $\displaystyle{ \mathrm{C}_{\mathrm{c}}(X) }$ of all continuous complex-valued functions of compact support on a locally compact Hausdorff space $\displaystyle{ X. }$ Consider a Borel regular measure $\displaystyle{ \mu }$ on $\displaystyle{ X, }$ and a functional $\displaystyle{ \psi }$ defined by $\displaystyle{ \psi(f) = \int_X f(x) d \mu(x) \quad \text{ for all } f \in \mathrm{C}_{\mathrm{c}}(X). }$ 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 $\displaystyle{ M }$ be a C*-algebra (more generally, an operator system in a C*-algebra $\displaystyle{ A }$) with identity $\displaystyle{ 1. }$ Let $\displaystyle{ M^+ }$ denote the set of positive elements in $\displaystyle{ M. }$

A linear functional $\displaystyle{ \rho }$ on $\displaystyle{ M }$ is said to be positive if $\displaystyle{ \rho(a) \geq 0, }$ for all $\displaystyle{ a \in M^+. }$

Theorem. A linear functional $\displaystyle{ \rho }$ on $\displaystyle{ M }$ is positive if and only if $\displaystyle{ \rho }$ is bounded and $\displaystyle{ \|\rho\| = \rho(1). }$[2]

### Cauchy–Schwarz inequality

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

## Applications to economics

Given a space $\displaystyle{ C }$, a price system can be viewed as a continuous, positive, linear functional on $\displaystyle{ C }$.