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
[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 $m^dagger$?
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