Saturated actions by finite-dimensional Hopf *-algebras on C*-algebras

If a finite group action alpha on a unital C*-algebra M is saturated, the canonical conditional expectation E : M -> M-alpha onto the fixed point algebra is known to be of index finite type with Index(E) = vertical bar G vertical bar in the sense of Watatani. More generally, if a finite-dimensional Hopf *-algebra A acts on M and the action is saturated, the same is true with Index(E) = dim(A). In this paper, we prove that the converse is true. Especially in case M is a commutative C*-algebra C(X) and alpha is a finite group action, we give an equivalent condition in order that the expectation E : C(X) -> C(X)(alpha) is of index finite type, from which we obtain that a is saturated if and only if G acts freely on X. Actions by compact groups are also considered to show that the gauge action gamma on a graph C*-algebra C*(E) associated with a locally finite directed graph E is saturated.