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

Abstract

If a finite group action $\alpha$ on a unital $C^{\ast }$-algebra $M$ is saturated, the canonical conditional expectation $E:M\to M^{\alpha }$ 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 $\ast$-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 $C^{\ast }$-algebra $C(X)$ and $\alpha$ is a finite group action, we give an equivalent condition in order that the expectation $E:C(X)\to C(X{)}^{\alpha }$ is of index finite type, from which we obtain that $\alpha$ 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 $C^{\ast }$-algebra $C^{\ast }(E)$ associated with a locally finite directed graph $E$ is saturated.

18 pages, to be published in Intern. J. Math