Algebra and Combinatorics Seminar

Date: October 8, 2021

Time: 3:00PM - 4:00PM

Location: BLOC 302

Speaker: Roberto Palomares, TAMU

Title: Q-systems and higher unitary idempotent completion for C* 2-categories

Abstract: A Q-system is a unitary version of a Frobenius algebra object in a tensor category or a C* 2-category. Q-systems were introduced by Longo to characterize the canonical endomorphism of a finite index inclusion of infinite von Neumann factors. Following work of Douglass-Reutter, a Q-system is also a unitary version of a higher idempotent. We will define a higher unitary idempotent completion for C* 2-categories called Q-system completion, and describe some of its properties and examples. We will show that C*Alg, the C* 2-category of right correspondences of unital C*-algebras is Q-system complete by adapting a technique from subfactors theory called realization. This result allows for the straightforward adaptation of subfactor results to C*-algebras, characterizing finite Jones-Watatani-index extensions of unital C*-algebras $A \subset B$ equipped with a faithful conditional expectation $E:B \to A$ in terms of the Q-systems in C*Alg.