For finite-dimensional (non-weak) Hopf C\*-algebras it is known that the antipode is always involutive, as claimed e.g. in <https://arxiv.org/pdf/1007.5283.pdf>. I couldn't find the same statement for weak Hopf C\*-algebras, however. Are there counterexamples?



Are there examples of finite-dimensional weak Hopf C*-algebras with non-involutive antipode?
According to [this paper](https://www.sciencedirect.com/science/article/pii/S002212369993522X) of Nikshych-Vainerman, a finite-dimensional weak Kac algebra is precisely a finite-dimensional [weak Hopf](https://en.wikipedia.org/wiki/Weak_Hopf_algebra) ${\rm C}^{\star}$-algebra with an involutive antipode ($S^2 = id$), and they give (p306) an explicit example of finite-dimensional weak Hopf [${\rm C}^{\star}$-algebra](https://en.wikipedia.org/wiki/C*-algebra) which is not a weak Kac algebra.


More generally, any finite index depth $2$ unital inclusion of [type ${\rm II}\_1$ factors](https://en.wikipedia.org/wiki/Von_Neumann_algebra#Type_II_factors) is characterized by a finite-dimensional weak Hopf ${\rm C}^{\star}$-algebra and in the weak Kac case, the index is an integer. So every non-integer finite index depth $2$ [subfactor](https://en.wikipedia.org/wiki/Subfactor) provides an example of finite-dimensional weak Hopf ${\rm C}^{\star}$-algebra with a non-involutive antipode. For any depth $n$ subfactor $(N \subset M)$ of finite index $|M:N|$, if $$N \subset M \subset M\_1 \subset M\_2 \subset M\_3 \subset \cdots $$ is the [Jones tower](https://en.wikipedia.org/wiki/Subfactor#Jones_tower), then $(N \subset M\_{n-2})$ is a depth $2$ subfactor of index $|M:N|^{n-1}$. The depth $n$ Temperley-Lieb-Jones subfactor has index $4cos^2(\frac{\pi}{n+1})$, and $[4cos^2(\frac{\pi}{n+1})]^{n-1}$ is not an integer for $n \neq 2,4$, so the corresponding weak Hopf ${\rm C}^{\star}$-algebra has a non-involutive antipode (it should also be the case for $n=4$, but not for $n=2$). The example of Nikshych-Vainerman above is $n=3$.




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[This paper](https://www.sciencedirect.com/science/article/pii/S0022123604000795) of Das proves by [planar algebra](https://en.wikipedia.org/wiki/Planar_algebra) techniques that for any finite index depth $2$ subfactor $(N \subset M)$, the relative commutants $N' \cap M\_1$ and $M' \cap M\_2$ has a structure of finite-dimensional weak Hopf ${\rm C}^{\star}$-algebra. What does the diagram on p84 say about the antipode $S$?


