In mathematics, an ordered algebra is an algebra over the real numbers R {\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 R {\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 σ ( A ∗ , A ) {\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 C R ( X ) {\displaystyle C_{\mathbb {R} }(X)} . If in addition every positive sequence of type l1 in A is order summable then A together with the Minkowski functional pe is isomorphic to the Banach algebra C R ( X ) {\displaystyle C_{\mathbb {R} }(X)} .
See also Ordered vector space Riesz space
References
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.
