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 $(c'^{\gamma}_{\alpha \beta})$ 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 $b_\alpha*b_\beta= \sum_\gamma {c'}_{\alpha\beta}^\gamma b_\gamma$.

Question: Is $c'^{\gamma}_{\alpha \beta} = c^{\gamma}_{\alpha \beta}$?
(Up to a permutation of the indices, a conjugation and a multiplicative constant).
If so, what's the exact formula and the proof?

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 p9).

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 for a proof in the non-weak case.

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 p9).

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 $a * b = \mathcal{F}(\mathcal{F}^{-1}(a).\mathcal{F}^{-1}(b))$.
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 $(c'^{\gamma}_{\alpha \beta})$ 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 $b_\alpha*b_\beta= \sum_\gamma {c'}_{\alpha\beta}^\gamma b_\gamma$.

Question: Is $c'^{\gamma}_{\alpha \beta} = c^{\gamma}_{\alpha \beta}$?
(Up to a permutation of the indices, a conjugation and a multiplicative constant).
If so, what's the exact formula and the proof?

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 p9).

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 $(c'^{\gamma}_{\alpha \beta})$ 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 $b_\alpha*b_\beta= \sum_\gamma {c'}_{\alpha\beta}^\gamma b_\gamma$.

Question: Is $c'^{\gamma}_{\alpha \beta} = c^{\gamma}_{\alpha \beta}$?
(Up to a permutation of the indices, a conjugation and a multiplicative constant).
If so, what's the exact formula and the proof?

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 p9).

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?

Edit (sept. 19, 2014)

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 a finite dimensional weak Kac algebra.
If we consider the finite index depth $2$ irreducible 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 p9).

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 $a * b = \mathcal{F}(\mathcal{F}^{-1}(a).\mathcal{F}^{-1}(b))$.
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 $(c'^{\gamma}_{\alpha \beta})$ 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 $b_\alpha*b_\beta= \sum_\gamma {c'}_{\alpha\beta}^\gamma b_\gamma$.

Question: Is $c'^{\gamma}_{\alpha \beta} = c^{\gamma}_{\alpha \beta}$?
(Up to a permutation of the indices, a conjugation and a multiplicative constant).
If so, what's the exact formula and the proof?

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 p9).