Ergodic Actions of Universal Quantum Groups on Operator Algebras

Abstract:

We construct ergodic actions of compact quantum groups on C ^*-algebras and von Neumann algebras, and exhibit phenomena of such actions that are of different nature from ergodic actions of compact groups.

In particular, we construct: (1) an ergodic action of the compact quantum A _u (Q) on the type III_λ Powers factor R _λ for an appropriate positive Q∈GL(2, ℝ); (2) an ergodic action of the compact quantum group A _u (n) on the hyperfinite II₁ factor R; (3) an ergodic action of the compact quantum group A _u (Q) on the Cuntz algebra for each positive matrix Q∈GL(n, ℂ); (4) ergodic actions of compact quantum groups on their homogeneous spaces, as well as an example of a non-homogeneous classical space that admits an ergodic action of a compact quantum group.