Suppose that $H$ is a Hopf algebra with normalised invariant integral (appropriate side) $\int:H\to \mathbb{C}$. The $H$ right comodule algebra $P$ is a Hopf Galois extension, so the canonical map $P\otimes\_A P\to P\otimes H$ is a 1-1 correspondence, where $A$ is the $H$ invariant part of $P$. 


There is an averaging map $E:P\to A$ given by $E(p)=p\_{(0)} \ \int(p\_{(1)})$ which gives a projection to the subalgebra $A$ and is an $A$ bimodule map.


Now let us also suppose that $P$ is a $C^\*$ algebra, and that $H$ is a Hopf star algebra with whatever other properties we need, and that the integral preserves positivity. Just when is $E$ a conditional expectation for $C^\*$ algebras, i.e. when does it preserve positivity? Any nice sufficient conditions would be interesting. 


Apologies if I have missed something obvious in the literature... 



Hopf Galois extensions and conditional expectations for C* algebras
First off, as you can see, the definition of $E$ doesn't require the Hopf-Galois condition. The positivity is a consequence of that for slicing by states. If $\phi$ is a state on a C$^\*$-algebra $C$, the map $\iota\otimes\phi\colon B \otimes\_{\rm min} C \to B$ characterized by $b \otimes c \mapsto \phi(c) b$ is (completely) positive, as the GNS representation for $\phi$ provides a Stinespring factorization of $\iota\otimes\phi$. Now, with $\phi = \int$ being the Haar state of a compact quantum group, $E$ can be presented as the composition of $(\iota \otimes \phi)$ and the coaction $\*$-homomorphism $P \to P \otimes\_{\rm min} H$ which is (completely) positive. So $E$ is (completely) positive as expected for a conditional expectation.


