2015/10/31 by Cyril Houdayer, Yusuke Isono
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Ergodic theory #Free group #Group (periodic table) #Measure (data warehouse) #Probability measure #Random Matrices and Applications #Second-countable space #Standard probability space #Subalgebra #Type (biology) #Von Neumann algebra #math.DS #math.GR #math.OA #msc:37A20 #msc:46L06 #msc:46L10 #msc:46L36
paper · pdf · doi:10.1007/s00220-016-2634-7
published as Comm. Math. Phys. 348 (2016), 991-1015 · 22 pages. v2: Final version
openalex publication_date 2016/05/10 · openalex created_date 2016/06/24 · arxiv created 2016/10/07 · arxiv updated 2016/11/03 · openalex updated_date 2026/08/05
We investigate the asymptotic structure of (possibly type III) crossed product von Neumann algebras M = B \rtimes Γ arising from arbitrary actions Γ\curvearrowright 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' ∩ Mω of any nonamenable von Neumann subalgebra with normal expectation N ⊂ M. We use this result to show that for any strongly ergodic essentially free nonsingular action Γ\curvearrowright (X, μ) of any bi-exact countable discrete group on a standard probability space, the corresponding group measure space factor \rm L^∞(X) \rtimes Γ has no nontrivial central sequence. Using recent results of Boutonnet-Ioana-Salehi Golsefidy [BISG15], we construct, for every 0 < λ≤ 1, a type IIIλ strongly ergodic essentially free nonsingular action \mathbf F_∞ \curvearrowright (Xλ, μλ) of the free group \mathbf F_∞ on a standard probability space so that the corresponding group measure space type IIIλ factor \rm L^∞(Xλ, μλ) \rtimes \mathbf F_∞ has no nontrivial central sequence by our main result. In particular, we obtain the first examples of group measure space type III factors with no nontrivial central sequence.