Let $\mathcal{A}$ be a finite dimensional $\mathbb{C}^* $-Hopf algebra. Let $B(\mathcal{A})$ be a basis of $\mathcal{A}$ and let $B(\mathcal{A}^* )=\{ \delta_x\in\mathcal{A}^* | x\in B(\mathcal{A}) \}$ be the corresponding dual basis of the dual Hopf algebra $\mathcal{A}^* $ satisfying $\delta_x(y)=\delta_{x,y}$. The canonical element of $\mathcal{A}\otimes\mathcal{A}^* $ defined as $$ \mathbf{c}=\sum_{x\in B(\mathcal{A})} x\otimes \delta_x\in \mathcal{A}\otimes\mathcal{A}^* . $$ It is straightforward to show that $\mathbf{c}$ is an invertible element of $\mathcal{A}\otimes\mathcal{A}^* $ and its inverse is given by $$\mathbf{c}^{-1}=(S\otimes \mathrm{id})\mathbf{c}=\sum_{x\in B(\mathcal{A})} S(x)\otimes \delta_x,$$ where $\mathrm{id}(x):=x$ is the identity map. The antipode axiom directly leads to $$\mathbf{c}^{-1}\cdot\mathbf{c}=\mathbf{c}\cdot\mathbf{c}^{-1}=1\otimes \epsilon,$$ where $1\otimes \epsilon$ is the unit element of $\mathcal{A}\otimes\mathcal{A}^* $.

Furthermore, the $* $-structure in $\mathcal{A}$ implies that $\mathbf{c}^{-1}=\mathbf{c}^* $, i.e., $\mathbf{c}$ is a unitary element of the algebra $\mathcal{A}\otimes\mathcal{A}^* $.

**Conjecture: for any finite dimensional $\mathbb{C}^* $-Hopf algebra $\mathcal{A}$, $\mathbf{c}$ has finite order in $\mathcal{A}\otimes\mathcal{A}^* $**, i.e., there exists a positive integer $n$ such that
$$\mathbf{c}^{n}=1\otimes \epsilon,$$
or equivalently, using Sweedler notation,
$$
\sum_{(z)} z_{(1)}z_{(2)}\ldots z_{(n)}=\epsilon(z)1,\quad\forall z\in \mathcal{A}.
$$

This conjecture is easily verified to be true for the group algebra $\mathbb{C}[G]$ of any finite group $G$, since there must exist an integer $n$ such that $g^n=1,~\forall g\in G$. I also verified this conjecture computationally in several $\mathbb{C}^* $-Hopf algebras of small dimension. **But is it true in general and how to prove it?**