• DOI:10.1007/s00220-016-2634-7
• Corpus ID: 119310275

Bi-Exact Groups, Strongly Ergodic Actions and Group Measure Space Type III Factors with No Central Sequence

@article{Houdayer2015BiExactGS,
  title={Bi-Exact Groups, Strongly Ergodic Actions and Group Measure Space Type III Factors with No Central Sequence},
  author={Cyril Houdayer and Yusuke Isono},
  journal={Communications in Mathematical Physics},
  year={2015},
  volume={348},
  pages={991-1015}
}

• Cyril Houdayer, Yusuke Isono
• Published 27 October 2015
• Mathematics
• Communications in Mathematical Physics

AbstractWe investigate the asymptotic structure of (possibly type III) crossed product von Neumann algebras $${M = B \rtimes \Gamma}$$M=B⋊Γ arising from arbitrary actions $${\Gamma \curvearrowright B}$$Γ↷B of bi-exact discrete groups (e.g. free groups) on amenable von Neumann algebras. We prove a spectral gap rigidity result for the central sequence algebra $${N' \cap M^\omega}$$N′∩Mω of any nonamenable von Neumann subalgebra with normal expectation $${N \subset M}$$N⊂M. We use this result to show that for any strongly ergodic essentially free nonsingular action $${\Gamma \curvearrowright (X, \mu)}$$Γ↷(X,μ) of any bi-exact countable discrete group on a standard probability space, the corresponding group measure space factor $${L^\infty(X) \rtimes \Gamma}$$L∞(X)⋊Γ has no nontrivial central sequence. Using recent results of Boutonnet et al. (Local spectral gap in simple Lie groups and applications, 2015), we construct, for every $${0 < \lambda \leq 1}$$0<λ≤1, a type $${{\rm III_\lambda}}$$IIIλ strongly ergodic essentially free nonsingular action $${\mathbf{F}_\infty \curvearrowright
(X_\lambda, \mu_\lambda)}$$F∞↷(Xλ,μλ) of the free group $${{\mathbf{F}}_\infty}$$F∞ on a standard probability space so that the corresponding group measure space type $${{\rm III_\lambda}}$$IIIλ factor $${L^\infty(X_\lambda, \mu_\lambda) \rtimes \mathbf{F}_\infty}$$L∞(Xλ,μλ)⋊F∞ has no nontrivial central sequence by our main result. In particular, we obtain the first examples of group measure space type $${{\rm III}}$$III factors with no nontrivial central sequence.