SATURATED ACTIONS BY FINITE-DIMENSIONAL HOPF *-ALGEBRAS ON C*-ALGEBRAS

If a finite group action α on a unital C*-algebra M is saturated, the canonical conditional expectation E : M → Mα onto the fixed point algebra is known to be of index finite type with Index (E) = |G| 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 α is a finite group action, we give an equivalent condition in order that the expectation E : C(X) → C(X)α is of index finite type, from which we obtain that α is saturated if and only if G acts freely on X. Actions by compact groups are also considered to show that the gauge action γ on a graph C*-algebra C*(E) associated with a locally finite directed graph E is saturated.