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 ${\displaystyle 0\in X}$ such that ${\displaystyle \operatorname {Re} f}$ is bounded above on ${\displaystyle M\cap \left(U-C\right)}$.

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 ${\displaystyle 0\in X}$ such that ${\displaystyle \operatorname {Re} f}$ is bounded above on ${\displaystyle M\cap \left(W-C\right)}$.

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

${\displaystyle \psi (f)=\int _{X}f(x)d\mu (x)\quad }$

for all ${\displaystyle f}$ in ${\displaystyle \mathrm {C} _{\mathrm {c} }(X)}$. 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 ${\displaystyle \left\|\rho \right\|=\rho (1)}$.[2]

Cauchy–Schwarz inequality

If ρ 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

${\displaystyle \left|\rho (b^{\ast }a)\right|^{2}\leq \rho (a^{\ast }a)\cdot \rho (b^{\ast }b).}$

See also

• Positive element
• Positive linear operator

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.
• 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 (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.