Ordered algebra explained

In mathematics, an ordered algebra is an algebra over the real numbers

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 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

\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

C _{R} (X) . 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 C _{R} (X) .

See also

• Ordered vector space
• Riesz space

Sources

• Book: Helmut H. Schaefer

. Schaefer . Helmut H. . Helmut H. Schaefer . Wolff . Manfred P. . Topological Vector Spaces . Springer New York Imprint Springer . . 3 . New York, NY . 1999 . 978-1-4612-7155-0 . 840278135 .