Bi-exact groups, strongly ergodic actions and group measure space type III factors with no central sequence

Abstract

We investigate the asymptotic structure of (possibly type III) crossed product von Neumann algebras $M = B \rtimes \Gamma$ arising from arbitrary actions $\Gamma \curvearrowright B$ 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 $N' \cap M^\omega$ of any nonamenable von Neumann subalgebra with normal expectation $N \subset M$. We use this result to show that for any strongly ergodic essentially free nonsingular action $\Gamma \curvearrowright (X, \mu)$ of any bi-exact countable discrete group on a standard probability space, the corresponding group measure space factor ${\rm L}^\infty(X) \rtimes \Gamma$ has no nontrivial central sequence. Using recent results of Boutonnet-Ioana-Salehi Golsefidy [BISG15], we construct, for every $0 < \lambda \leq 1$, a type III$_\lambda$ strongly ergodic essentially free nonsingular action $\mathbf F_\infty \curvearrowright (X_\lambda, \mu_\lambda)$ of the free group $\mathbf F_\infty$ on a standard probability space so that the corresponding group measure space type III$_\lambda$ factor ${\rm L}^\infty(X_\lambda, \mu_\lambda) \rtimes \mathbf F_\infty$ 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.