I recently came across Kac algebra. They are roughly Hopf algebras and $C^\*$-algebras with compatible structures. It follows from Artin–Wedderburn theorem that every semisimple complex Hopf algebra can be seen as a $C^\*$-algebra structure, but I am unsure whether this structure will be compatible with the comultiplication. I guess that this is not the case, as I think that there are dimensions where Kac algebras have been classified, but semisimple Hopf algebras have not.


**Question:** Is every semisimple/finite-dimensional complex Hopf algebra a Kac algebra?


Something similar has been previously asked [here](https://mathoverflow.net/questions/324111/is-there-a-non-kac-complex-finite-dimensional-semisimple-hopf-algebra), but the question is unanswered. If this is unknown in general, it would also be helpful to know if this holds under some stronger assumptions (eg. for a given dimension).


EDIT: Lemma IV.8.2 in *Quantum Groups* by Kassel could be useful:


A Hopf algebra $H$ has a Hopf $\*$-algebra structure if and only if there exists an antilinear automorphism $\gamma$ of $H$ such that


$(i)$ the map $\gamma$ is a morphism of real algebras and an antimorphism of real coalgebras, and


$(ii)$ we have $\gamma^2 = (S \gamma)^2 = \text{id}\_H$.



Hopf algebras vs. Kac algebras
As pointed out in the comments all semisimple Hopf algebras of dimension up to $23$ are Kac algebras. So it seems like dimension $24$ is the smallest one where this is open.


Corollary 9.7 of "Weakly group-theoretical and solvable fusion categories" by Etingof, Nikshych and Ostrik says that a semisimple Hopf algebra of dimension $p q^2$ is either a Kac algebra or a twisted group algebra (by a twist corresponding to the subgroup $(Z/qZ)^2$) or the dual of a twisted group algebra.


