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Some Topics in Infinite Dimensional Algebra

Bossaller, Daniel P.
http://orcid.org/0000-0001-7176-8998
http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1520332321386827

Abstract Details

2018, Doctor of Philosophy (PhD), Ohio University, Mathematics (Arts and Sciences).
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 such as the R-algebra of infinite matrices with finitely many nonzero entries, and the column-finite, and row-and-column finite matrix algebras. The fourth chapter is work towards determining when Leavitt path algebras have the invertibility and local invertibility properties. In [28–30], the authors completely classify when the Leavitt path algebra arising from a finite graph is invertible. I follow up on these author’s research in two ways. First, using the author’s analysis of “source loops,” I show that so long as the finite graph has no non-isolated source loops, the Leavitt path algebra is locally invertible. Secondly I give various conditions under which certain elements of Leavitt path algebras are locally invertible. Then I give some graph theoretic characterizations which are sufficient for local invertibility of the Leavitt path algebra. Among other characterizations it is shown that every von Neumann regular Leavitt path algebra and every directly finite Leavitt path algebra is locally invertible. Finally in the fifth chapter I will outline some avenues for future research on the problems described above, in many cases giving partial results towards the resolutions of the questions.
Sergio Lopez-Permouth (Advisor)
89 p.
Mathematics
Associativity; Infinite Matrices; Bases; Strongly Regular Elements; Leavitt Path Algebras

Recommended Citations

Citations

  • Bossaller, D. P. (2018). Some Topics in Infinite Dimensional Algebra [Doctoral dissertation, Ohio University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1520332321386827

    APA Style (7th edition)

  • Bossaller, Daniel. Some Topics in Infinite Dimensional Algebra. 2018. Ohio University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1520332321386827.

    MLA Style (8th edition)

  • Bossaller, Daniel. "Some Topics in Infinite Dimensional Algebra." Doctoral dissertation, Ohio University, 2018. http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1520332321386827

    Chicago Manual of Style (17th edition)

ohiou1520332321386827
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This open access ETD is published by Ohio University and OhioLINK.