Lucas Hall

Research

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

In more than five words, I work with Operator Algebras, one branch in the Functional 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 actions 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 beyond 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., 115 (2025), no.1, Paper no. 2, 24pp.
Abstract

The Stone–von Neumann theorem is a fundamental result which unified the competing quantum-mechanical models of matrix mechanics and wave mechanics. In this article, we continue the broad generalization set out by Huang and Ismert and by Hall, Huang, and Quigg, analyzing representations of locally compact quantum-dynamical systems defined on Hilbert modules, of which the classical result is a special case. We introduce a pair of modular representations which subsume numerous models available in the literature and, using the classical strategy of Rieffel, prove a Stone–von Neumann-type theorem for maximal actions of regular locally compact quantum groups on elementary C*-algebras. In particular, we generalize the Mackey–Stone–von Neumann theorem to regular locally compact quantum groups whose trivial actions on the complex numbers are maximal and recover the multiplicity results of Hall, Huang, and Quigg. With this characterization in hand, we prove our main result showing that if a quantum dynamical system satisfies the multiplicity assumption of the generalized Stone–von Neumann theorem, and if the coefficient algebra admits a faithful state, then the spectrum of the iterated crossed product consists of a single point. In the case of a separable coefficient algebra or a regular acting quantum group, we further characterize features of this system, and thus obtain a partial converse to the Stone–von Neumann theorem in the quantum group setting. As a corollary, we show that a regular locally compact quantum group satisfies the generalized Stone–von Neumann theorem if and only if it is strongly regular.

Work in Progress

Grants, Fellowships, and Awards

Invited Talks

Contributed Talks

Local Talks

See also the service seminars which I have organized.

External Links:

Service

"Operators are best studied in community."-- G. Yu

I hold myself responsible for ensuring the field of operator algebras, and mathematics generally, is accessible to any whom find interest in it, as well as inspiring future generations in the pursuit of math research. I believe strongly in active participation in local seminars, equipping students and peers with tools and strategies to develop their mathematical talent, and the power of an invitation.

Past Service Programs

• Panelist: MSU New Postdoc Orientation Panel
• Commuting Operators Seminar. University of Haifa (nationally broadcasted), April 2024
• Dilation Theory Learning Seminar. University of Haifa (globally broadcasted) November 2023
• Free Probability Book Club. Michigan State University (interstate broadcasted) November 2023
• Panelist: AMS-MSU Postdoc Ask Me Anything Panel. MSU Fall 2023
• Co-Organizer: Travel Grants for YMC*A. MSU Spring 2023
• Co-Organizer: East Coast Operator Algebras Symposium. MSU Fall 2022
• Senior Teaching Assistant: SoMSS TA Training. ASU Summer 2021
• Coordinator/Lead Facilitator: Respect is a Part of Academics Workshop. ASU Spring 2021
• SoMSS Peer Mentoring Program. ASU Fall 2020