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 ${\displaystyle v\geq 0,}$ ${\displaystyle f(v)}$ 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 ${\displaystyle C}$.

See also

• Positive element – Group with a compatible partial orderPages displaying short descriptions of redirect targets
• Positive linear operator – Concept in functional analysis

References

1. Schaefer & Wolff 1999, pp. 225–229.
2. Murphy, Gerard. "3.3.4". C*-Algebras and Operator Theory (1st ed.). Academic Press, Inc. p. 89. ISBN 978-0125113601.

Bibliography

• Kadison, Richard, Fundamentals of the Theory of Operator Algebras, Vol. I : Elementary Theory, American Mathematical Society. ISBN 978-0821808191.
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• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.