Analysis Seminar

Roberto Hernandez-Palomares, Texas A&M University

"Q-systems and higher unitary idempotent completion for C*-algebras"

Q-systems were introduced by Longo to study finite index inclusions of infinite von Neumann factors. A Q-system is a unitary version of a Frobenius algebra object in a tensor category or a C* 2-category. By work of Müger, Q-systems give an axiomatization of the standard invariant of a finite index subfactor.

Following work of Douglass-Reutter, a Q-system is also a unitary version of a higher idempotent. In this talk, we will describe a higher unitary idempotent completion for C* 2-categories called Q-system completion.

Our main goal is to show that C*Alg, the C* 2-category of right correspondences of unital C*-algebras is Q-system complete. To do so, we will use the graphical calculus for C* 2-categories, and adapt a subfactor reconstruction technique called realization, which is inverse to Q-system completion. This result allows for the straightforward adaptation of subfactor results to C*-algebras, characterizing finite index extensions of unital C*-algebras equipped with a faithful conditional expectation in terms of the Q-systems in C*Alg. If time allows, we will discuss an application to induce new symmetries of C*-algebras from old via Q-system completion. This is joint work with Chen, C. Jones and Penneys (arXiv: 2105.12010).

Zoom link: https://uwaterloo.zoom.us/j/94186354814?pwd=NGpLM3B4eWNZckd1aTROcmRreW96QT09