Some Topics in Infinite Dimensional Algebra
Daniel P. Bossaller · OhioLink ETD Center (Ohio Library and Information Network) · 2018
The following dissertation is a combination of three papers, emanating from two separate and mostly disjoint projects.The first project is an investigation of when the product of arbitrary infinite matrices is defined and associative.In the case where infinite matrices represent endomorphisms on an infinite-dimensional vector space, this question is trivial since multiplication of these matrices amounts to composition of the endomorphisms, a well-defined and associative operation.In the case of infinite matrices in general, this question becomes very nontrivial.In Chapter 2 we will explore this question of associativity and give necessary and sufficient conditions for a set of three infinite matrices A, B, and C to have an associative product.Afterwards we will give some applications for this characterization.The second project which makes up this dissertation is an exploration of R-algebras which have the property that there exists a basis B for the algebra which consists solely of strongly regular elements.This work was originally inspired by [26] and [27] which investigated algebras which have bases consisting solely of invertible elements.In the third chapter, it is shown that these strongly regular elements are also satisfy a slightly weaker version of invertibility which is dubbed "local invertibility."Because local invertibility does not presuppose the existence of a multiplicative identity, we may examine algebras which fail to have a unit.Among algebras which satisfy the local invertibility property are all finite dimensional unital algebrs over a division ring D, and many examples of infinite matrix algebras Thanks goes first and foremost to Sergio López-Permouth, who has patiently mentored me throughout my graduate education at OU. Thank you for having always having an alternate perspective or interesting questions regarding my research.Thanks especially for those conversations which made me into a better mathematician and, more importantly, a better colleague.Second I want to thank my parents, Kim and Andy Bossaller.The dedication you showed to my education throughout my life contributed in no small part to the completion of this degree.