Let $\mathbb{A}$ be a finite  dim. **weak**  Hopf $C^*$-algebra (or semisimple finite quantum groupoid) 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 algebra, $\mathbb{A}$ is isomorphic to a direct sum of matrix algebras. Let $B = \{b_1, \dots , b_n  \}$ a matrix basis.      

Let the structure constants $(c^{\gamma}_{\alpha \beta})$ and $(d^{\gamma}_{\alpha \beta})$  of the comultiplication $\Delta$ and the convolution product $*$ given by $\Delta(b_{\gamma}) = \sum_{\alpha \beta} c^{\gamma}_{\alpha \beta} . b_{\alpha} \otimes b_{\beta}$ and $b_\alpha*b_\beta= \sum_\gamma {d}_{\alpha\beta}^\gamma b_\gamma$.   



**Question**: Is $d^{\gamma}_{\alpha \beta} = \delta \overline{c^{\gamma}_{\alpha \beta}}$?   
[with $\delta = dim(\mathbb{A})^{1/2}$]  
  
*Remark*: See proposition 8.16 p40 of [this paper][1] for a proof in the non-weak case. 
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There is the following definition of the Fourier transform in the planar algebra framework, *but I guess there is an equivalent definition in the Hopf algebra framework*.   
 
A finite dimensional weak Hopf $C^*$algebra is by definition a finite dimensional weak Kac algebra.  
If we consider the finite index depth $2$ subfactor planar algebra $\mathcal{P} = \mathcal{P}(R \subset R\rtimes \mathbb{A})$, with $R$ the hyperfinite ${\rm II}_1$ factor, then $\mathbb{A}$ and $\hat{\mathbb{A}}$ are the $2$-boxes algebras $\mathcal{P}_{2,+}$ and $\mathcal{P}_{2,-}$.  
The Fourier transform $\mathcal{F} : \mathcal{P}_{2,\pm} \to \mathcal{P}_{2,\mp}$ is the $1$-click rotation (see for example [here][2] p9).


  [1]: http://arxiv.org/pdf/1505.06649v7.pdf
  [2]: http://arxiv.org/pdf/1408.1165v1.pdf