Positive linear functional

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

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 [math]\displaystyle{ V }[/math] is a complex vector space, it is assumed that for all [math]\displaystyle{ v\ge0, }[/math] [math]\displaystyle{ f(v) }[/math] is real. As in the case when [math]\displaystyle{ V }[/math] is a C*-algebra with its partially ordered subspace of self-adjoint elements, sometimes a partial order is placed on only a subspace [math]\displaystyle{ W\subseteq V, }[/math] and the partial order does not extend to all of [math]\displaystyle{ V, }[/math] in which case the positive elements of [math]\displaystyle{ V }[/math] are the positive elements of [math]\displaystyle{ W, }[/math] by abuse of notation. This implies that for a C*-algebra, a positive linear functional sends any [math]\displaystyle{ x \in V }[/math] equal to [math]\displaystyle{ s^{\ast}s }[/math] for some [math]\displaystyle{ s \in V }[/math] to a real number, which is equal to its complex conjugate, and therefore all positive linear functionals preserve the self-adjointness of such [math]\displaystyle{ x. }[/math] 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 [math]\displaystyle{ X }[/math] be an Ordered topological vector space with positive cone [math]\displaystyle{ C \subseteq X }[/math] and let [math]\displaystyle{ \mathcal{B} \subseteq \mathcal{P}(X) }[/math] denote the family of all bounded subsets of [math]\displaystyle{ X. }[/math] Then each of the following conditions is sufficient to guarantee that every positive linear functional on [math]\displaystyle{ X }[/math] is continuous:

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

Continuous positive extensions

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

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

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

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

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

Examples

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

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

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

Cauchy–Schwarz inequality

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

Applications to economics

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

See also

• Positive linear operator

Bibliography

• Kadison, Richard, Fundamentals of the Theory of Operator Algebras, Vol. I : Elementary Theory, American Mathematical Society. ISBN:978-0821808191.
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. 
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. 
• Trèves, François (August 6, 2006). Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.