Positive linear functional

In mathematics, more specifically in functional analysis, a positive linear functional on an ordered vector space <math>(V, \leq)</math> is a linear functional <math>f</math> on <math>V</math> so that for all positive elements <math>v \in V,</math> that is <math>v \geq 0,</math> it holds that <math display=block>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>V</math> is a complex vector space, it is assumed that for all <math>v\ge0,</math> <math>f(v)</math> is real. As in the case when <math>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>W\subseteq V,</math> and the partial order does not extend to all of <math>V,</math> in which case the positive elements of <math>V</math> are the positive elements of <math>W,</math> by abuse of notation. This implies that for a C*-algebra, a positive linear functional sends any <math>x \in V</math> equal to <math>s^{\ast}s</math> for some <math>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>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 functionalsEdit

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

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

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

Continuous positive extensionsEdit

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

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

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

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

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

ExamplesEdit

Consider, as an example of <math>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>\mathrm{C}_{\mathrm{c}}(X)</math> of all continuous complex-valued functions of compact support on a locally compact Hausdorff space <math>X.</math> Consider a Borel regular measure <math>\mu</math> on <math>X,</math> and a functional <math>\psi</math> defined by <math display=block>\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)Edit

Let <math>M</math> be a C*-algebra (more generally, an operator system in a C*-algebra <math>A</math>) with identity <math>1.</math> Let <math>M^+</math> denote the set of positive elements in <math>M.</math>

A linear functional <math>\rho</math> on <math>M</math> is said to be Template:Em if <math>\rho(a) \geq 0,</math> for all <math>a \in M^+.</math>

Theorem. A linear functional <math>\rho</math> on <math>M</math> is positive if and only if <math>\rho</math> is bounded and <math>\|\rho\| = \rho(1).</math><ref name=Murphy>Template:Cite book</ref>

Cauchy–Schwarz inequalityEdit

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