Properly Proximal von Neumann Algebras

Abstract

We introduce the notion of proper proximality for finite von Neumann algebras, which naturally extends the notion of proper proximality for groups. Apart from the group von Neumann algebras of properly proximal groups, we provide a number of additional examples, including examples in the settings of free products, crossed products, and compact quantum groups. Using this notion, we answer a question of Popa by showing that the group von Neumann algebra of a nonamenable inner amenable group cannot embed into a free group factor. We also introduce a notion of proper proximality for probability measure preserving actions, which gives an invariant for the orbit equivalence relation. This gives a new approach for establishing strong ergodicity type properties, and we use this in the setting of Gaussian actions to expand on solid ergodicity results first established by Chifan and Ioana, and later generalized by Boutonnet. The techniques developed also allow us to answer a problem left open by Anantharaman-Delaroche in 1995, by showing the equivalence between the Haagerup property and the compact approximation property for II1 factors.

Measure equivalence rigidity via s-malleable deformations

“…Since any non-amenable property Gamma von Neumann algebra cannot embed into the free group factor L(F n ) (see [Oza04]), Popa asked in [Pop21] if it still true that the group von Neumann algebra of a non-amenable inner amenable group cannot embed into L(F n ). Recently, inspired by the notion of properly proximal groups [BIP21] (see also [IPR19]), Ding, Kunnawalkam Elayavalli, and Peterson developed in [DKP22] subtle boundary techniques to define a notion of proper proximality for tracial von Neumann algebras, and as a consequence, they answered Popa's question in a positive way. As a particular case of Theorem E, we give a new proof for Popa's question by using methods from Popa's deformation/rigidity theory.…”

“…Consequently, we derive several orbit equivalence rigidity results for actions of product groups that belong to M. Finally, for groups Γ satisfying condition (ii), we show that all embeddings of group von Neumann algebras of non-amenable inner amenable groups into L(Γ) are 'rigid'. In particular, we provide an alternative solution to a question of Popa that was recently answered by Ding, Kunnawalkam Elayavalli, and Peterson [Properly Proximal von Neumann Algebras, Preprint (2022), arXiv:2204.00517].…”

On Divided-Type Connectivity of Graphs

“…The notion of Haagerup property is firstly introduced for locally compact groups in [Ha79] and then extended to tracial von Neumann algebras in [Ch83]. Following [BF11], [OOT17] and [DKP22], a II 1 factor M has the Haagerup property if it admits a mixing M -bimodule which has almost central unit vectors. 2.7.…”