TITLE: Are there examples of finite-dimensional weak Hopf C*-algebras with non-involutive antipode?
QUESTION [5 upvotes]: 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?

REPLY [2 votes]: This is an answer to the question in the OPs answer. 
I just took a brief look into

Rehren - Weak C Hopf symmetry* in Quantum Groups Symposium at
  "Group21", eds. H.-D. Doebner et al., Goslar 1996 Proceedings, Heron
  Press, Sofia (1997), pp. 62-69. https://arxiv.org/abs/q-alg/9611007

which does the type III case.
The point is that $S^{-1}(q)=S^\ast(q^\ast)$, which is what the diagram on p84 in the mentioned article says.