# Does every special $C^*$-Frobenius algebra have a unit?

MathOverflow Asked by quantumOrange on January 1, 2022

I have a rather basic question about $$C^*$$-Frobenius algebras (also called Q-systems). Any pointers or references will be most helpful!

We are given a finite-dimensional complex Hilbert space $$mathbb{V}$$ with a multiplication $$m: mathbb{V} otimes mathbb{V} rightarrow mathbb{V}$$ (a linear map) such that

1. $$m$$ is associative
2. $$m^dagger m = mathbb{I}$$ (the Identity on $$mathbb{V}$$), that is, $$m$$ is an isometry, and
3. $$m$$ and $$m^dagger$$ fulfill the Frobenius relation, namely, $$(m^dagger otimes mathbb{I}) (mathbb{I} otimes m) = m m^dagger$$,

[Here $$m^dagger$$ is the Hermitian conjugate (adjoint) of $$m$$. If $$m$$ is viewed as a matrix from $$mathbb{V} otimes mathbb{V}$$ to $$mathbb{V}$$ then $$m^dagger$$ is obtained by first transposing this matrix and then applying entry-wise complex conjugation.]

These relations are depicted by the string diagrams shown below:

My question is: Given this data, does the multiplication $$m$$ necessarily have a unit? If yes, can it be expressed in terms of $$m$$ and $$m^dagger$$?