Positive linear functional
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 is a complex vector space, it is assumed that for all , is real. As in the case when is a C*-algebra with its partially ordered subspace of self-adjoint elements, sometimes a partial order is placed on only a subspace , and the partial order does not extend to all of , in which case the positive elements of are the positive elements of , by abuse of notation.[clarification needed] This implies that for a C*-algebra, a positive linear functional sends any equal to for some to a real number, which is equal to its complex conjugate, and therefore all positive linear functionals preserve the self-adjointness of such . 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 be an ordered topological vector space with positive cone and let denote the family of all bounded subsets of . Then each of the following conditions is sufficient to guarantee that every positive linear functional on is continuous:
- has non-empty topological interior (in ).
- is complete and metrizable and .
- is bornological and is a semi-complete strict -cone in .
- is the inductive limit of a family of ordered Fréchet spaces with respect to a family of positive linear maps where for all , where is the positive cone of .
Continuous positive extensions
The following theorem is due to H. Bauer and independently, to Namioka.
- Theorem: Let be an ordered topological vector space (TVS) with positive cone , let be a vector subspace of , and let be a linear form on . Then has an extension to a continuous positive linear form on if and only if there exists some convex neighborhood of such that is bounded above on .
- Corollary: Let be an ordered topological vector space with positive cone , let be a vector subspace of . If contains an interior point of then every continuous positive linear form on has an extension to a continuous positive linear form on .
- Corollary: Let be an ordered vector space with positive cone , let be a vector subspace of , and let be a linear form on . Then has an extension to a positive linear form on if and only if there exists some convex absorbing subset in containing such that is bounded above on .
Proof: It suffices to endow with the finest locally convex topology making into a neighborhood of .
- Consider, as an example of , 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 of all continuous complex-valued functions of compact support on a locally compact Hausdorff space . Consider a Borel regular measure on , and a functional defined by
- for all in . 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 be a C*-algebra (more generally, an operator system in a C*-algebra ) with identity . Let denote the set of positive elements in .
A linear functional on is said to be positive if , for all .
- Theorem. A linear functional on is positive if and only if is bounded and .
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- Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
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