COVARIANT REPRESENTATIONS FOR COACTIONS OF HOPF ${\rm C}^*$-ALGEBRAS

Abstract

Given a coaction $\alpha$ of a Hopf ${\rm C}^*$-algebra $A$ on a ${\rm C}^*$-algebra $B$ with an $\alpha$-invariant ${\rm C}^*$-subalgebra $C$, and a conditional expectation $E:B \rightarrow C$ commuting with $\alpha$, it is shown that if $(\pi, u)$ is a covariant representation of the system ($C, A, \alpha\mid_C$), then there is an associated covariant representation ($\tilde{\pi}, \tilde{u}$) of the system ($B, A, \alpha$), where $\tilde{\pi}$ is the representation induced from $\pi$ up to $B$ via $E$, and $\tilde{u}$ is a unitary corepresentation of $A$ naturally associated with $u$. Some applications are also discussed, including a lifting of ergodic coactions to von Neumann algebras, and a characterization of the amenability of multiplicative unitary operators via infinite tensor product covariant representations.