Taen-Yu Dai

Publication Date

3-30-1971

Abstract

R. DeMarr (unpublished) has begun a study of Banach algebras as subalgebras of partially ordered linear algebras which are Dedekind a -complete. In [3] 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 a -complete. This leads us to study a more gen­eralized function algebra by using the order structure. In this paper from analytical point of view we will study some special classes of partially ordered linear algebras which are Dedekind a -complete. In chapter 1 we assume the algebra which has the property: If x >= 1, then x-1exists and x-1 > o. We will see that the algebra has this property is actually a function algebra and, hence, it has no non-zero nilpotents, and idempotents lie between O and 1.Moreover, it is an f-ring. In chapter 2, we will study the general structure of the algebra which has the special property: If x > 1, then x-1 exists and 1 > X -1. In chapter, we will discuss the algebra which is a lattice and has the special property given in chapter 2. Then in such algebra there always exists a non-trivial 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 Parts we will also discuss an algebra which has so called the Perron-Frobenius property [4].

Degree Name

Mathematics

Level of Degree

Doctoral

Department Name

Mathematics & Statistics

First Committee Member (Chair)

Ralph Elgin DeMarr

Second Committee Member

Bernard Epstein

Third Committee Member

Illegible

Language

English

Document Type

Dissertation

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