Amenable Ergodic Actions , Hyperfinite Factors , and Poincaré Flows

Turbulence, Representations, and Trace-preserving Actions

We establish criteria for turbulence in certain spaces of C∗-algebra representations and apply this to the problem of nonclassifiability by countable structures for group actions on a standard atomless probability space (X,μ) and on the hyperfinite II1 factor R. We also prove that the conjugacy action on the space of free actions of a countably infinite amenable group on R is turbulent, and tha...

. O A ] 1 3 A ug 2 00 8 TURBULENCE , REPRESENTATIONS , AND TRACE - PRESERVING ACTIONS

We establish criteria for turbulence in certain spaces of C *-algebra representations and apply this to the problem of nonclassifiability by countable structures for group actions on a standard atomless probability space (X, µ) and on the hyperfinite II1 factor R. We also prove that the conjugacy action on the space of free actions of a countably infinite amenable group on R is turbulent, and t...

Ergodic Theoretic Characterization of Left Amenable Lau Algebras

A Converse to Dye’s Theorem

Every non-amenable countable group induces orbit inequivalent ergodic equivalence relations on standard Borel probability spaces. Not every free, ergodic, measure preserving action of F2 on a standard Borel probability space is orbit equivalent to an action of a countable group on an inverse limit of finite spaces. There is a treeable non-hyperfinite Borel equivalence relation which is not univ...

Ergodic Actions of Universal Quantum Groups on Operator Algebras

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 Au(Q) on the type IIIλ Powers factor Rλ for an appropriate positive Q ∈ GL(2, R); (2). an ergodic action of the compact...