Positive linear functional

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In mathematics, more specifically in functional analysis, a positive linear functional on an ordered vector space is a linear functional on so that for all positive elements , that is , it holds that

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[edit]

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 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:

  1. has non-empty topological interior (in ).[1]
  2. is complete and metrizable and .[1]
  3. is bornological and is a semi-complete strict -cone in .[1]
  4. 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 .[1]

Continuous positive extensions[edit]

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

Theorem:[1] 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:[1] 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:[1] 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 .

Examples[edit]

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)[edit]

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 .[2]

Cauchy–Schwarz inequality[edit]

If ρ is a positive linear functional on a C*-algebra , then one may define a semidefinite sesquilinear form on by . Thus from the Cauchy–Schwarz inequality we have

See also[edit]

References[edit]

  1. ^ a b c d e f g h i j 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[edit]