The definition of unitary fusion category I just come across a definition of the unitary fusion category:
A fusion category $\mathcal{C}$ over the complex number is said to be unitary if we have:


*

*We have a Hilbert space structure on each Hom space, and we denote the
inner product by $\langle \cdot, \cdot \rangle$.

*We have a contravariant endofunctor * on $\mathcal{C}$ which is identity on objects.

*We have $\|fg\| \leq \|f\|\|g\|$ and $\|f^*f\| = \|f\|^2$ for 
each $f \in Hom(Y,Z)$ and $g \in Hom(X,Y)$.

*We have $(f \otimes g)^* = f^* \otimes g^*$ for any morphism $f$ and $g$. 

*All structure isomorphisms for simple objects are unitary.


I am confused about the notation in condition 4. Does $\|f\| = \sqrt{
\langle f, g\rangle}$ or $\|f\|$ means the operator norm? From my understanding, I guess $\|f\|$ in the definition should means the operator norm. Would anybody please clarify this for me? Thank you!
 A: Given a unitary fusion category, there is more than one natural norm on the (finite dimensional) hom-space $Hom(X,Y)$.
There is the operator norm:
That's the norm under which the unitary fusion category is a $C^*$-category.
Let's denote that norm $\|\,\,\|_\infty$.
That norm satisfies the $C^*$-identity $\|f^*f\|_\infty=\|f\|_\infty^2$.
And there's the Hilbert-Schmidt norm:
(That's the norm used by Baez et al in Higher-Dimensional Algebra II). Let's denote that norm by $\|\,\,\|_2$.
With respect to that norm, a morphism $f:X\to Y$ which factors through an irreducible object will satisfy $\|f\|_2^2 = \|f\|_\infty^2$.
Using the above formula, the Hilbert-Schmidt norm of an arbitrary morphism is then uniquely determined by means of the polarization identity.
Warning:There are more than one way to normalize the Hilbert-Schmidt norm, and so some care is needed when dealing with that notion. Given a morphism $f:X\to Y$ that factors through an irreducible object $Z$, one could also decide to let $\|f\|_2^2 = d_Z\cdot\|f\|_\infty^2$, where $d_Z$ is, e.g., the quantum dimension of the object $Z$.

Once the normalization of Hilbert-Schmidt norm has been fixed, then there is also the associated
Trace-class norm $\|\,\,\|_1$: It satisfies $\|f^*f\|_1=\|f\|_2^2$ for any morphism $f$. It also satisfies$\|pu\|_1=\|p\|_1$ for any positive morphism $p:Y\to Y$ and unitary morphism $u:X\to Y$.
