# Unitary structures on fusion categories

A unitary fusion category is a fusion category with a $C^*$-tensor structure. Hence, in principle, a fusion category could have more than one unitary structure. Does exist a fusion category with more than one (non equivalent) unitary structures.

Unitarity is equivalent to the existence of basis such that the $F$-matrices are unitary. This property is to be unitarizible. The question in matricial terms is: suppose that you have two different basis such that the $F$-matrices are unitary, Can be unitary gauged one basis in the other? or in oher words, there exist always unitary change of basis from one basis to the other basis?

Is every equivalence $F:\mathcal C\to \mathcal D$ between unitary fusion categories naturally equivalent to a unitary equivalence (i.e., one which is a dagger functor)
I see that Galindo-Hong-Rowell have some results in that direction (see $\S$5.5, on page 21 of his paper), but they don't exactly addresses this question.
I am curious about this, too. I guess there are also fusion categories without a $\mathrm{C}^\ast$-tensor structure, what goes wrong there? Do you happen to have a reference where the notion of a "unitary fusion category" is used (or even better) introduced? How would I call a MTC which is $\mathrm{C}^\ast$? – Marcel Bischoff May 26 '13 at 12:01
But isn't that just polar decomposition? That is to say, if you can conjugate one unitary by $X$ to get another, then you can also get between them by conjugating by the unitary matrix $X \sqrt{X X^\dagger}^{-1}$? – Noah Snyder Jun 13 '13 at 4:37