Misplaced Pages

This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these messages) This article is an orphan, as no other articles link to it. Please introduce links to this page from related articles; try the Find link tool for suggestions. (March 2024) This article relies largely or entirely on a single source. Relevant discussion may be found on the talk page. Please help improve this article by introducing citations to additional sources. Find sources: "Ordered algebra" – news · newspapers · books · scholar · JSTOR (June 2020) (Learn how and when to remove this message)

In mathematics, an ordered algebra is an algebra over the real numbers ${\displaystyle \mathbb {R} }$ with unit e together with an associated order such that e is positive (i.e. e ≥ 0), the product of any two positive elements is again positive, and when A is considered as a vector space over ${\displaystyle \mathbb {R} }$ then it is an Archimedean ordered vector space.

Properties

Let A be an ordered algebra with unit e and let C denote the cone in A (the algebraic dual of A) of all positive linear forms on A. If f is a linear form on A such that f(e) = 1 and f generates an extreme ray of C then f is a multiplicative homomorphism.

Results

Stone's Algebra Theorem: Let A be an ordered algebra with unit e such that e is an order unit in A, let A denote the algebraic dual of A, and let K be the ${\displaystyle \sigma \left(A^{*},A\right)}$-compact set of all multiplicative positive linear forms satisfying f(e) = 1. Then under the evaluation map, A is isomorphic to a dense subalgebra of ${\displaystyle C_{\mathbb {R} }(X)}$. If in addition every positive sequence of type l in A is order summable then A together with the Minkowski functional p_e is isomorphic to the Banach algebra ${\displaystyle C_{\mathbb {R} }(X)}$.

See also

• Ordered vector space
• Riesz space

References

1. ^ Schaefer & Wolff 1999, pp. 250–257.

Sources

• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 3. New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.

• Functional analysis