Positive linear functional

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. 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

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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

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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   in   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 the origin of   such that   is bounded above on  

Proof: It suffices to endow   with the finest locally convex topology making   into a neighborhood of  

Examples

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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   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)

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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

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If   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  

Applications to economics

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Given a space  , a price system can be viewed as a continuous, positive, linear functional on  .

See also

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References

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  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

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