*-Hopf algebroids

We introduce a theory of $*$-structures for bialgebroids and Hopf algebroids over a $*$-algebra, defined in such a way that the relevant category of (co)modules is a bar category. We show that if $H$ is a Hopf $*$-algebra then the action Hopf algebroid $A\# H$ associated to a braided-commutative algebra in the category of $H$-crossed modules is a full $*$-Hopf algebroid and the Ehresmann-Schauenburg Hopf algebroid $\mathcal{L}(P,H)$ associated to a Hopf-Galois extension or quantum group principal bundle $P$ with fibre $H$ forms a $*$-Hopf algebroid pair, when the relevant (co)action respects $*$. We also show that Ghobadi's bialgebroid associated to a $*$-differential structure $(\Omega^{1},\rm d)$ on $A$ forms a $*$-bialgebroid pair and its quotient in the pivotal case a $*$-Hopf algebroid pair when the pivotal structure is compatible with $*$. We show that when $\Omega^1$ is simultaneously free on both sides, Ghobadi's Hopf algebroid is isomorphic to $\mathcal{L}(A\#H,H)$ for a smash product by a certain Hopf algebra $H$.