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

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

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Quantum Groups Seminar [QGS]

Series comments: This online 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.

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