Let $\mathbb{A}$ be a finite dimensional weak Hopf $C^*$-algebra, and $\hat{\mathbb{A}}$ its dual.
Let the Fourier transform $\mathcal{F}: \mathbb{A} \to \hat{\mathbb{A}}$ and the convolution $a * b = \mathcal{F}^{-1}(\mathcal{F}(a).\mathcal{F}(b))$.
As an algebras, $\mathbb{A}$ is isomorphic to $\bigoplus_k M_{n_k}(\mathbb{C})$ and admits a matrix basis $B = \{b_1, \dots , b_n \}$.
(Note that $b_\alpha$ is a matrix $E_{ij}$, so that $tr(\vert b_{\alpha} \vert) = 1$)
The structure constants $(c^{\gamma}_{\alpha \beta})$ of $\Delta$ are given by $\Delta(b_{\gamma}) = \sum_{\alpha \beta} c^{\gamma}_{\alpha \beta} . b_{\alpha} \otimes b_{\beta}$

Question: Is there a formula for the convolution $b_{\alpha} * b_{\beta}$ in term of the structure constants?
If yes, what's this formula and what's the proof (or a reference).

Remark: my guess is that $b_\alpha*b_\beta= \frac{1}{\sqrt{n}} \sum_\gamma \bar{c}_{\alpha\beta}^\gamma b_\gamma$
If so, why not defining a weak Hopf $C^*$-algebra by using a convolution instead of a comultiplication?