Speaker
Lara Ismert, University of New Mexico
Abstract
In this talk, we recall the definition of a quantum graph and a quantum Cuntz—Krieger algebra due to Brannan, Eifler, Voigt, and Weber. In this picture, the vertex set is replaced by a quantum set consisting of a finite-dimensional C*-algebra and a state, and the simple graph’s adjacency matrix is generalized to a quantum Schur idempotent on the quantum set. From a quantum graph in this framework, the quantum Cuntz—Krieger algebra is constructed as the universal C*-algebra generated by a quantum Cuntz—Krieger family, which generalizes the relations on a classical adjacency matrix. In this paradigm, the vertices of a classical graph translate to direct summands of a quantum set, so in this talk, we call a quantum graph on a single full matrix algebra a single-vertex quantum graph. Even for a single-vertex graph, determining the isomorphism class of the quantum Cuntz—Krieger algebra remains challenging. However, the local quantum Cuntz—Krieger algebra, introduced in 2023 by Brannan, Hamidi, Ismert, Nelson, and Wasilewski, was shown to be isomorphic to a Cuntz—Pimsner algebra for a well-studied C*-correspondence arising from the quantum adjacency matrix. In this talk, we give a formal isomorphism with the rephrasing graph on M_2, and we discuss how the Cuntz—Pimsner algebras arising from single-vertex quantum graphs are in general Morita equivalent to Cuntz algebras (due to Marrero and Muhly). We conclude by proposing future directions for characterizing simplicity of local quantum Cuntz—Krieger algebras on general quantum graphs.
Lara Ismert, University of New Mexico
Abstract
In this talk, we recall the definition of a quantum graph and a quantum Cuntz—Krieger algebra due to Brannan, Eifler, Voigt, and Weber. In this picture, the vertex set is replaced by a quantum set consisting of a finite-dimensional C*-algebra and a state, and the simple graph’s adjacency matrix is generalized to a quantum Schur idempotent on the quantum set. From a quantum graph in this framework, the quantum Cuntz—Krieger algebra is constructed as the universal C*-algebra generated by a quantum Cuntz—Krieger family, which generalizes the relations on a classical adjacency matrix. In this paradigm, the vertices of a classical graph translate to direct summands of a quantum set, so in this talk, we call a quantum graph on a single full matrix algebra a single-vertex quantum graph. Even for a single-vertex graph, determining the isomorphism class of the quantum Cuntz—Krieger algebra remains challenging. However, the local quantum Cuntz—Krieger algebra, introduced in 2023 by Brannan, Hamidi, Ismert, Nelson, and Wasilewski, was shown to be isomorphic to a Cuntz—Pimsner algebra for a well-studied C*-correspondence arising from the quantum adjacency matrix. In this talk, we give a formal isomorphism with the rephrasing graph on M_2, and we discuss how the Cuntz—Pimsner algebras arising from single-vertex quantum graphs are in general Morita equivalent to Cuntz algebras (due to Marrero and Muhly). We conclude by proposing future directions for characterizing simplicity of local quantum Cuntz—Krieger algebras on general quantum graphs.