Graph \(C^*\)-algebras
Lecture, GOALS 2025, Virtual
I lectured on universal \(C^*\)-algebras and graph \(C^*\)-algebras. Here is a recording. I learned from last time not to trust zoom, and I used my own recording software this time.
Talks and presentations
Ideals, approximate units
Lecture, GOALS 2025, Virtual
I lectured on ideals and approximate units in \(C^*\)-algebras for GOALS 2025, which was held virtually. Here is a recording. Unfortunately zoom decided to cover half of the screen with the participants.
Ordinal graphs and their \(C^*\)-algebras
Contributed Talk, COSY 2025, Waterloo, Ontario, Canada
We introduce a class of left-cancellative categories which generalizes the category of paths of a directed graph by allowing paths to have ordinal length. We use generators and relations to study the Cuntz-Krieger algebra defined by Spielberg. For each ordinal we construct an associated \(C^*\)-correspondence. Then we apply Eryüzlü and Tomforde’s condition (S) and obtain a Cuntz-Krieger uniqueness theorem for ordinal graphs.
Ordinal graphs and their \(C^*\)-algebras
Contributed Lightning Talk, COMPhY 2025, Waterloo, Ontario, Canada
We introduce a class of left-cancellative categories which generalizes the category of paths of a directed graph by allowing paths to have ordinal length. We use generators and relations to study the Cuntz-Krieger algebra defined by Spielberg. For each ordinal we construct an associated \(C^*\)-correspondence. Then we apply Eryüzlü and Tomforde’s condition (S) and obtain a Cuntz-Krieger uniqueness theorem for ordinal graphs.
Ordinal graphs and their \(C^*\)-algebras
Invited Talk, MAA Rocky Mountain Section 2025 Meeting, Boulder, Colorado, USA
We introduce a class of left-cancellative categories which generalizes the category of paths of a directed graph by allowing paths to have ordinal length. We use generators and relations to study the Cuntz-Krieger algebra defined by Spielberg. For each ordinal we construct an associated \(C^*\)-correspondence. Then we apply Eryüzlü and Tomforde’s condition (S) and obtain a Cuntz-Krieger uniqueness theorem for ordinal graphs.
Ordinal graphs and their \(C^*\)-algebras
Invited Talk, IMPAN North Atlantic Noncommutative Geometry Seminar, Virtual
We introduce a class of left-cancellative categories which generalizes the category of paths of a directed graph by allowing paths to have ordinal length. We use generators and relations to study the Cuntz-Krieger algebra defined by Spielberg. For each ordinal we construct an associated \(C^*\)-correspondence. Then we apply Eryüzlü and Tomforde’s condition (S) and obtain a Cuntz-Krieger uniqueness theorem for ordinal graphs.
Ordinal graphs and their \(C^*\)-algebras
Contributed Talk, ASU,Embry-Riddle \(C^*\) Seminar, Tempe, AZ, USA
We introduce a class of left-cancellative categories which generalizes the category of paths of a directed graph by allowing paths to have ordinal length. We use generators and relations to study the Cuntz-Krieger algebra defined by Spielberg. For each ordinal we construct an associated \(C^*\)-correspondence. Then we apply Eryüzlü and Tomforde’s condition (S) and obtain a Cuntz-Krieger uniqueness theorem for ordinal graphs.
Ordinal graphs and their \(C^*\)-algebras
Contributed Talk, GOALS seminar, Virtual
We introduce a class of left-cancellative categories which generalizes the category of paths of a directed graph by allowing paths to have ordinal length. We use generators and relations to study the Cuntz-Krieger algebra defined by Spielberg. For each ordinal we construct an associated \(C^*\)-correspondence. Then we apply Eryüzlü and Tomforde’s condition (S) and obtain a Cuntz-Krieger uniqueness theorem for ordinal graphs.
A Cuntz-Krieger Uniqueness Theorem for a Class of Left-Cancellative Categories
Contributed Talk, ASU,Embry-Riddle \(C^*\) Seminar, Tempe, AZ, USA
We introduce a class of left-cancellative small categories for which there is a Cuntz-Krieger uniqueness theorem. The categories arise naturally as a generalization of directed graphs for which paths are allowed to have lengths which are ordinals. We will describe a collection of \(C^*\)-correspondences which generalize the usual correspondence for a directed graph. Under suitable conditions, these correspondences satisfy condition (S) of Eryüzlü and Tomforde, and we describe how a transfinite induction argument yields the uniqueness theorem.
Topological Quiver \(C^*\)-Algebras & Group Actions
Contributed Talk, GPOTS 2024, University of Nebraska-Lincoln, Lincoln, NE, USA
Topological quivers are a somewhat natural generalization of topological graphs, where we insist that edge space is additionally endowed with a family of Radon measures, indexed by the vertex space, satisfying some regularity conditions. In this series of talks, we aim to present the needed background material to understand what a topological quiver \(C^*\)-algebra is, as well as discuss some results we hope to generalize from Deaconu, Kumjian and Quigg in the topological graph setting. One of these main results being that a locally compact group acting freely and properly on a topological quiver induces a Morita equivalence between the reduced crossed product of the quiver \(C^*\)-algebra and the \(C^*\)-algebra of the corresponding ‘quotient quiver’.
How Quantum Mechanics Motivates \(C^*\)-algebras
Contributed Talk, ASU,Embry-Riddle \(C^*\) Seminar, Tempe, AZ, USA
In this talk we introduce Quantum mechanics from first principles. We illustrate how Stone’s theorem and the Stone-von Neumann theorem bridge the gap between the canonical commutation relations for unbounded operators and the Weyl relations for unitary groups. Time permitting, we’ll recast the Stone-von Neumann theorem in terms of \(C^*\) crossed products and discuss how this motivates natural generalizations of the Stone-von Neumann theorem.
Topological Quiver \(C^*\)-Algebras & Group Actions
Contributed Talk, ASU,Embry-Riddle \(C^*\) Seminar, Tempe, AZ, USA
Topological quivers are a somewhat natural generalization of topological graphs, where we insist that edge space is additionally endowed with a family of Radon measures, indexed by the vertex space, satisfying some regularity conditions. In this series of talks, we aim to present the needed background material to understand what a topological quiver \(C^*\)-algebra is, as well as discuss some results we hope to generalize from Deaconu, Kumjian and Quigg in the topological graph setting. One of these main results being that a locally compact group acting freely and properly on a topological quiver induces a Morita equivalence between the reduced crossed product of the quiver \(C^*\)-algebra and the \(C^*\)-algebra of the corresponding ‘quotient quiver’.