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 $M=B⋊\Gamma$ 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 $N^{\prime }\cap M^{\omega }$ 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 ${\mathrm{L}}^{\infty }(X)⋊\Gamma$ has no nontrivial central sequence. Using recent results of Boutonnet-Ioana-Salehi Golsefidy [BISG15], we construct, for every $0<\lambda \le 1$, 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 ${\mathrm{L}}^{\infty }(X_{\lambda },{\mu }_{\lambda })⋊{\mathbf{F}}_{\infty }$ 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.

22 pages. v2: Final version

References in corpus (2)

• Asymptotic structure of free Araki-Woods factors
• A characterization of fullness of continuous cores of type III${}_1$ free product factors

Cited by in corpus (9)

• Strongly ergodic equivalence relations: spectral gap and type III invariants
• A remark on fullness of some group measure space von Neumann algebras
• Nonsingular Bernoulli actions of arbitrary Krieger type
• Solidity of type III Bernoulli crossed products
• Structure of modular invariant subalgebras in free Araki-Woods factors
• Factoriality, Connes' type III invariants and fullness of amalgamated free product von Neumann algebras
• Boundary and rigidity of nonsingular Bernoulli actions
• Fullness of crossed products of factors by discrete groups
• Matrix representations of multidimensional integral and ergodic operators