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

Roberto Hernández Palomares (Texas A&M University, USA)

27-Jun-2022, 14:00-15:00 (21 months ago)

Abstract: 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 the 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 Q. Chen, C. Jones and D. Penneys (arXiv: 2105.12010).

category theoryoperator algebrasquantum algebra

Audience: researchers in the topic


Quantum Groups Seminar [QGS]

Series comments: This seminar aims to bring together experts in the area of quantum groups. The seminar topics will cover the theory of quantum groups and related structures in a large sense: Hopf algebras, operator algebras, q-deformations, higher categories and related branches of noncommutative mathematics.

The zoom link will be distributed by mail, so please join the mailing list if you are interested in attending the seminar.

Organizers: Rubén Martos, Frank Taipe*, Makoto Yamashita
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