TITLE: Does every Frobenius algebra in a monoidal *-category give a Q-system?
QUESTION [13 upvotes]: Suppose that C is a fusion C*-cateogry and that A is an irreducible Frobenius algebra object in C, is there always a Frobenius algebra A' isomorphic to A such that A' is a Longo Q-system (that is the coproduct on A' is an isometry)?  In other words, wlog can one assume that the coproduct is the * of the product?
The motivation of this question is to understand whether the theory of irreducible Frobenius algebra objects in monoidal *-categories actually agrees on the nose with the theory of irreducible subfactors or whether there's a small loophole.

REPLY [2 votes]: This is not a full answer. The answer is yes for weakly group-theoretical fusion categories. The question is equivalent to: let  C be a unitary fusion category,  does every indecomposable C-module category admit a compatible unitary structure (see GMR, for all definitions). In  Theorem 5.20, we prove that a weakly group-theoretical fusion category admits a unique unitary structure and every indecomposable module category also admits a unique compatible unitary structure.