Ordered algebra

In mathematics, an ordered algebra is an algebra over the real numbers with unit e together with an associated order such that e is positive, the product of any two positive elements is again positive, and when A is considered as a vector space over 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^* of all positive linear forms on A.
If f is a linear form on A such that f = 1 and f generates an extreme ray of C^* then f is a multiplicative homomorphism.

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 [weak-* topology|]-compact set of all multiplicative positive linear forms satisfying f = 1. Then under the evaluation map, A is isomorphic to a dense subalgebra of. 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.