Lucas Hall

In five words, I study linear algebra on infinite dimensions.

In more than five words, I work with Operator Algebras, one of the newer branches in the Analysis limb of Mathematics research. Broadly construed, I am interested in studying symmetries and dynamics of these operator algebras, which has come to include actions of groups and groupoids as well as coactions of locally compact quantum groups on associated operator algebras. One instructive lens through which I view these dynamics includes C*-correspondences as well as various operator algebras which one may build from them. I am always looking for new connections to fields of research outside of my immediate expertise.

Publications

1. Groupoid Semidirect Product Fell Bundles I- Actions by Isomorphisms, J. Operator Theory, 89 (2023) no.1: 125-153
Abstract

Given an action of a groupoid by isomorphisms on a Fell bundle (over another groupoid), we form a semidirect-product Fell bundle, and prove that its C*-algebra is isomorphic to a crossed product.

2. Groupoid Semidirect Product Fell Bundles II- Principal Actions and Stabilization, Indiana Univ. Math. J., 72 (2023), no. 3, 1147-1173
Abstract

Given a free and proper action of a groupoid on a Fell bundle (over another groupoid), we give an equivalence between the semidirect-product and the generalized-fixed-point Fell bundles, generalizing an earlier result where the action was by a group. As an application, we show that the Stabilization Theorem for Fell bundles over groupoids is essentially another form of crossed-product duality.

3. The Modular Stone-von Neumann Theorem, J. Operator Theory, 89 (2023), no.2, 571-586
Abstract

In this paper, we use the tools of nonabelian duality to formulate and prove a far-reaching generalization of the Stone-von Neumann Theorem to modular representations of actions and coactions of locally compact groups on elementary C*-algebras. This greatly extends the Covariant Stone-von Neumann Theorem for Actions of Abelian Groups recently proven by L. Ismert and the second author. Our approach is based on a new result about Hilbert C*-modules that is simple to state yet is widely applicable and can be used to streamline many previous arguments, so it represents an improvement -- in terms of both efficiency and generality -- in a long line of results in this area of mathematical physics that goes back to J. von Neumann's proof of the classical Stone-von Neumann Theorem.

4. Coactions and Skew Products for Topological Quivers, Proc. Edinburgh Math. Soc. II, 67 (2024), no. 3, 921-946
Abstract

Given a cocycle on a topological quiver by a locally compact group, the author constructs a skew product topological quiver, and determines conditions under which a topological quiver can be identified as a skew product. We investigate the relationship between the C*-algebra of the skew product and a certain native coaction on the C*-algebra of the original quiver, finding that the crossed product by the coaction is isomorphic to the skew product. As an application, we show that the reduced crossed product by the dual action is Morita equivalent to the C*-algebra of the original quiver.

5. The Covariant Stone-von Neumann Theorem for Locally Compact Quantum Groups, Lett. Math. Phys., Accepted 2024
Abstract

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Grants and Fellowships

• Zuckerman Postdoctoral Scholar, University of Haifa, 2023
• Conference: Young Mathematicians in C*-Algebras 2023, DMS-2247448
• University Graduate Fellowship, Arizona State University, 2019
• Walter E. Koss Endowed Fellowship, Texas A&M UNiversity, 2016
• New American University Scholar, Arizona State University, 2012
• National Hispanic Scholar, College Board, 2011

Invited Talks

• "The Covariant Stone von-Neumann Theorem for Locally Compact Quantum Groups," JMM Special Session: Research Presentations by Math Alliance Scholar Doctorates, II. Seattle, WA. January 2025
• "TBD," Ben Gurion University of the Negev, Israel.
• "Quantifying Proximity of Almost Commuting Unitaries to Bonafide Commuting Unitaries," Israeli NonCommutative Analysis Seminar. The Technion, Israel. November 2024
• "Topological Classification of Skew Products for Quiver C*-Algebras," North Atlantic Noncommutative Geometry Seminar. Virtual Broadcast. April 2024
• "Topological Classification of Some Coactions on C*-algebras of Topological Quivers," Quantum Symmetries Workshop. IMPAN, Poland. February 2024.
• "A Survey on Quantum Groups," ASUERAU C*-algebras Seminar. Arizona State University, August 2023.
• "The Quantum Covariant Stone-von Neumann Theorem," YMC*A. KU Leuven BE, August 2023.
• "The Modular Stone-von Neumann Theorem," Wabash Mini-Conference. IUPUI, Indianapolis, November 2023
• "Skew Products for Topological Quivers," Joint Math Meetings Special Session 57A. Virtual, April 2022
• "Coactions You Can See," UCSD Functional Analysis Seminar. University of California San Diego, January 2022.

Contributed Talks

• "Skewing Constructions for Topological Quivers," Groundwork in Operator Algebras Learning Seminar. Virtual, November 2023.
• "Coactions and Skew Products for Topological Quivers," YMC*A. WWU M\"unster, August 2021.
• "Skew Products for Topological Quivers," Summer Schools in Operator Algebras. Virtual, June 2021.
• "Semidirect Product Fell Bundles (Expanded)," Groundwork in Operator Algebras Learning Seminar. Virtual, December 2020.
• "Semidirect Product Fell Bundles," Canadian Operator Symposium. Virtual, 2020
• "The Union of Frame Theoretic and Gegenbauer Reprojection Techniques to Improve Image Construction Rates," AMS Session on Functional Analysis. Joint Math Meetings, January 2015.

Local Talks

See also the service seminars which I have organized.

• "Arveson's Extension Theorems" Haifa U. Dilation Theory Learning Seminar, March 2024.
• "Some Coactions on Topological Quiver C*-algebras" UHaifa-Techninon Noncommutative Analysis Seminar, January 2024.
• "Dilation Theorems" Haifa U. Dilation Theory Learning Seminar, January 2024.
• "Further Results on CP Maps" Haifa U. Dilation Theory Learning Seminar, January 2024.
• "Techniques and Consequences for Positive Maps," Haifa U. Dilation Theory Learning Seminar, December 2023.
• "Introduction to Dilation Theory," Haifa U. Dilation Theory Learning Seminar, November 2023.
• "Crossed Product C*-algebras," MSU OARS Seminar, November 2023.
• "Quantum Dynamics and the Covariant Stone-von Neumann Theorem," MSU MPOA Seminar, October 2023.
• "More on Quantum Groups: Corepresentation Theory," MSU OARS Seminar, September 2023.
• "A Survey on Quantum Groups," MSU OARS Seminar, September 2023.
• "Approximately Finite Dimensional C*-algebras," MSU OARS Seminar, January 2023.
• "Graph Algebras I and II," MSU OARS Seminar, November 2022.
• "Skew Products- Coactions You Can See," MSU MPOA Seminar, September 2022.
• "Stone-von Neumann Theorems," ASUERAU C*-Seminar, February 2022.
• "Coactions and Topological Quivers," ASUERAU C*-Seminar, November 2021.
• "An Exposition on the Hao-Ng Problem," ASUERAU C*-Seminar, September 2020.
• "Primer: Groupoids and Their C*-algebras," ASU C*-Seminar, February 2020.

External Links:

MathSciNet

orchid

arXiv

Zuckerman Institute

Math Geneology Project