Ordered Algebra

In mathematics, an ordered algebra is an algebra over the real numbers [math]\displaystyle{ \mathbb{R} }[/math] 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 [math]\displaystyle{ \mathbb{R} }[/math] 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.^[1]

Results

Stone's Algebra Theorem:^[1] 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 [math]\displaystyle{ \sigma\left( A^{*}, A \right) }[/math]-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 [math]\displaystyle{ C_{\mathbb{R}}(X) }[/math]. If in addition every positive sequence of type l¹ in A is order summable then A together with the Minkowski functional p_e is isomorphic to the Banach algebra [math]\displaystyle{ C_{\mathbb{R}}(X) }[/math].

See also

• Ordered vector space
• Riesz space

References

Sources

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

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