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On Graph Algebras

Schenkel, Timothy L.
http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1650622144443202

Abstract Details

2022, Doctor of Philosophy (PhD), Ohio University, Mathematics (Arts and Sciences).
The work presented here aims to contribute to the study of graph algebras. We concentrate our work to studying the regular ideals, and introducing graph algebras associated to infinite-rank graphs. We first give a vertex set description for the gauge-invariant regular ideals of locally convex, row-finite k-graph C∗-algebras. We also show that Condition (B) is preserved when taking the quotient by a gauge-invariant, regular ideal. Then we show that when a locally-convex row-finite k-graph satisfies Condition (B), all regular ideals are gauge-invariant. Secondly, we generalize the notion of a k-graph into (countable) infinite rank. We then define our C∗-algebra in a similar way as in k-graph C∗-algebras. With this construction we are able to find analogues to the Gauge Invariant Uniqueness and Cuntz-Krieger Uniqueness Theorems. We also show that the N-graph C∗-algebras can be viewed as the inductive limit of k-graph C∗-algebras. This gives a nice way to describe the gauge-invariant ideal structure. Additionally, we describe the vertex-set for regular gauge-invariant ideals of our N-graph C∗-algebras. Lastly, we give analogous theorems to our C∗-algebra case for both. In particular we examine the basic, graded, regular ideals for a locally-convex, row-finite Kumjian-Pask algebra and find similar results to that of k-graph C∗-algebras. We then take our construction of the N-graph into the algebraic setting and receive many similarities to the C∗-algebra construction.
Adam Fuller (Advisor)
Kenneth Hicks (Committee Member)
Marcel Bischoff (Committee Member)
Sergio Lopez-Permouth (Committee Member)
80 p.
Mathematics
Operator Theory; Rings and Algebras; Graph Algebras

Recommended Citations

Citations

  • Schenkel, T. L. (2022). On Graph Algebras [Doctoral dissertation, Ohio University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1650622144443202

    APA Style (7th edition)

  • Schenkel, Timothy. On Graph Algebras. 2022. Ohio University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1650622144443202.

    MLA Style (8th edition)

  • Schenkel, Timothy. "On Graph Algebras." Doctoral dissertation, Ohio University, 2022. http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1650622144443202

    Chicago Manual of Style (17th edition)

ohiou1650622144443202
105
© 2022, all rights reserved.
This open access ETD is published by Ohio University and OhioLINK.