On some special classes of partially ordered linear algebras

Abstract

R. DeMarr (unpublished) has begun a study of Banach algebras as sub-algebras of partially ordered linear algebras, which are Dedekind σ complete. He has shown that the real Banach algebra of norm-bounded linear operators (mapping a real Banach space into itself) can be made into a partially ordered linear algebra which is Dedekind σ complete. This leads us to study a more generalized function algebra by using the order structure. In this paper, from an analytical point of view, we study some special classes of partially ordered linear algebras which are Dedekind σ complete. In Section I we assume the algebra which has the property: If x ⩾ 1, then x −1 exists and x −1 ⩾ 0. We see that the algebra which has this property is actually a function algebra and, hence, it has no nonzero nilpotents, and idempotents lie between 0 and 1. Moreover, it is an f ring. In Section II, we study the general structure of the algebra which has the special property: If x ⩾ 1, then x −1 exists and 1 ⩾ x −1. In Section III, we discuss the algebra which is a lattice and has the special property given in Section II. Then in such algebra there always exists a nontrivial multiplicative linear function mapping the algebra into itself. By using this function we can study some of the properties of the algebra. In all three sections we also discuss an algebra which has the so-called Perron-Frobenius property.