Positive linear functional

In mathematics, more specifically in functional analysis, a positive linear functional on an ordered vector space (V, ≤) is a linear functional f on V so that for all positive elements v of V, that is v≥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 V is a complex vector space, it is assumed that for all v≥0, f(v) is real. As in the case when 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 of V, and the partial order does not extend to all of V, in which case the positive elements of V are the positive elements of W, by abuse of notation. This implies that for a C*-algebra, a positive linear functional sends any x in V equal to s*s for some 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. 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 X be an ordered topological vector space with positive cone C and let ${\displaystyle {\mathcal {B}}}$ denote the family of all bounded subsets of X. Then each of the following conditions is sufficient to guarantee that every positive linear functional on X is continuous:

1. C has non-empty topological interior (in X).[1]
2. X is complete and metrizable and X = C - C.[1]
3. X is bornological and C is a semi-complete strict ${\displaystyle {\mathcal {B}}}$-cone in X.[1]
4. 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 X be an ordered topological vector space (TVS) with positive cone C, let M be a vector subspace of E, and let f be a linear form on M. Then f has an extension to a continuous positive linear form on X if and only if there exists some convex neighborhood U of 0 in X such that ${\displaystyle \operatorname {Re} f}$ is bounded above on ${\displaystyle M\cap \left(U-C\right)}$.

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

Corollary:[1] Let X be an ordered vector space with positive cone C, let M be a vector subspace of E, and let f be a linear form on M. Then f has an extension to a positive linear form on X if and only if there exists some convex absorbing subset W in X containing 0 such that ${\displaystyle \operatorname {Re} f}$ is bounded above on ${\displaystyle M\cap \left(W-C\right)}$.

proof: It suffices to endow X with the finest locally convex topology making W into a neighborhood of 0.

Examples

• Consider, as an example of 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) of all continuous complex-valued functions of compact support on a locally compact Hausdorff space X. Consider a Borel regular measure μ on X, and a functional ψ defined by

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

for all f in C_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 M be a C*-algebra (more generally, an operator system in a C*-algebra A) with identity 1. Let M⁺ denote the set of positive elements in M.

A linear functional ρ on M is said to be positive if ρ(a) ≥ 0, for all a in M⁺.

Theorem. A linear functional ρ on M is positive if and only if ρ is bounded and ||ρ||=ρ(1).[2]

Cauchy–Schwarz inequality

If ρ is a positive linear functional on a C*-algebra A, then one may define a semidefinite sesquilinear form on A by <a, b> := ρ(b^*a). Thus from the Cauchy–Schwarz inequality we have

${\displaystyle |\rho (b^{*}a)|^{2}\leq \rho (a^{*}a)\rho (b^{*}b).}$

See also

• Positive element
• Positive linear operator

References

1. Schaefer 1999, p. 225-229. sfn error: no target: CITEREFSchaefer1999 (help)
2. Murphy, Gerard. "3.3.4". C*-Algebras and Operator Theory (1st ed.). Academic Press, Inc. p. 89. ISBN 978-0125113601.

• 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.CS1 maint: ref=harv (link)
• Trèves, François (August 6, 2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.CS1 maint: ref=harv (link) CS1 maint: date and year (link)