Positive linear functional

In mathematics, more specifically in functional analysis, a positive linear functional on an ordered vector space ( V , ≤ ) {\displaystyle (V,\leq )} is a linear functional f {\displaystyle f} on V {\displaystyle V} so that for all positive elements v ∈ V , {\displaystyle v\in V,} that is v ≥ 0 , {\displaystyle v\geq 0,} it holds that f ( v ) ≥ 0. {\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 V {\displaystyle V} is a complex vector space, it is assumed that for all v ≥ 0 , {\displaystyle v\geq 0,} f ( v ) {\displaystyle f(v)} is real. As in the case when V {\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 W ⊆ V , {\displaystyle W\subseteq V,} and the partial order does not extend to all of V , {\displaystyle V,} in which case the positive elements of V {\displaystyle V} are the positive elements of W , {\displaystyle W,} by abuse of notation. This implies that for a C*-algebra, a positive linear functional sends any x ∈ V {\displaystyle x\in V} equal to s ∗ s {\displaystyle s^{\ast }s} for some s ∈ V {\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 x . {\displaystyle x.} This property is exploited in the GNS construction to relate positive linear functionals on a C*-algebra to inner products.

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

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

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

Theorem:^[1] Let X {\displaystyle X} be an ordered topological vector space (TVS) with positive cone C , {\displaystyle C,} let M {\displaystyle M} be a vector subspace of E , {\displaystyle E,} and let f {\displaystyle f} be a linear form on M . {\displaystyle M.} Then f {\displaystyle f} has an extension to a continuous positive linear form on X {\displaystyle X} if and only if there exists some convex neighborhood U {\displaystyle U} of 0 {\displaystyle 0} in X {\displaystyle X} such that Re ⁡ f {\displaystyle \operatorname {Re} f} is bounded above on M ∩ ( U − C ) . {\displaystyle M\cap (U-C).}

Corollary:^[1] Let X {\displaystyle X} be an ordered topological vector space with positive cone C , {\displaystyle C,} let M {\displaystyle M} be a vector subspace of E . {\displaystyle E.} If C ∩ M {\displaystyle C\cap M} contains an interior point of C {\displaystyle C} then every continuous positive linear form on M {\displaystyle M} has an extension to a continuous positive linear form on X . {\displaystyle X.}

Corollary:^[1] Let X {\displaystyle X} be an ordered vector space with positive cone C , {\displaystyle C,} let M {\displaystyle M} be a vector subspace of E , {\displaystyle E,} and let f {\displaystyle f} be a linear form on M . {\displaystyle M.} Then f {\displaystyle f} has an extension to a positive linear form on X {\displaystyle X} if and only if there exists some convex absorbing subset W {\displaystyle W} in X {\displaystyle X} containing the origin of X {\displaystyle X} such that Re ⁡ f {\displaystyle \operatorname {Re} f} is bounded above on M ∩ ( W − C ) . {\displaystyle M\cap (W-C).}

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

Consider, as an example of V , {\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 C c ( X ) {\displaystyle \mathrm {C} _{\mathrm {c} }(X)} of all continuous complex-valued functions of compact support on a locally compact Hausdorff space X . {\displaystyle X.} Consider a Borel regular measure μ {\displaystyle \mu } on X , {\displaystyle X,} and a functional ψ {\displaystyle \psi } defined by ψ ( f ) = ∫ X f ( x ) d μ ( x )  for all  f ∈ C c ( X ) . {\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.

Let M {\displaystyle M} be a C*-algebra (more generally, an operator system in a C*-algebra A {\displaystyle A} ) with identity 1. {\displaystyle 1.} Let M + {\displaystyle M^{+}} denote the set of positive elements in M . {\displaystyle M.}

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

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

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

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

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

• 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. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.