  We construct ergodic actions of compact quantum groups on C^*-algebras and
von Neumann algebras, and exhibit phenomena of such actions that are of a
different nature from ergodic actions of compact Lie groups.
  In particular, we construct: (1). ergodic actions of the compact quantum
groups $A_u(Q)$ on the Powers factors; (2). ergodic actions of the compact
quantum groups $A_u(n)$ on the hyperfinite II_1 factor; (3). ergodic actions of
the compact quantum groups $A_u(Q)$ on the Cuntz algebras; (4). ergodic actions
of general compact quantum groups on their homogeneous spaces and an example of
a non-homogeneous classical space that nevertheless admits an ergodic action of
a compact quantum group.
