# Von Neumann equivalence and properly proximal groups

@article{Ishan2019VonNE, title={Von Neumann equivalence and properly proximal groups}, author={Ishan Ishan and Jesse Peterson and Lauren C. Ruth}, journal={arXiv: Operator Algebras}, year={2019} }

We introduce a new equivalence relation on groups, which we call von Neumann equivalence, that is coarser than both measure equivalence and $W^*$-equivalence. We introduce a general procedure for inducing actions in this setting and use this to show that many analytic properties, such as amenability, property (T), and the Haagerup property, are preserved under von Neumann equivalence. We also show that proper proximality, which was defined recently by Boutonnet, Ioana, and the second author…

## 10 Citations

### Von Neumann equivalence and group approximation properties

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The notion of von Neumann equivalence (vNE), which encapsulates both measure equivalence and W ∗-equivalence, was introduced recently in [IPR19], where it was shown that many analytic properties,…

### Properly Proximal von Neumann Algebras

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. 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…

### An upgrading theorem for properly proximal von Neumann algebras

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- 2022

Using computations in the bidual of $\mathbb{B}(L^2M)$ we develop a technique at the von Neumann algebra level to upgrade relative proper proximality to full proper proximality in the presence of…

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- 2021

In this paper, the notion of proper proximality (introduced in [BIP18]) is studied and classified in various families of groups. We show that if a group acts non-elementarily by isometries on a tree…

### Von Neumann equivalence and $M_d$ type approximation properties

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- 2022

. We show that M d -approximation-property, M d -weak-amenability, and M d -weak-Haagerup-property are invariant for von Neumann equivalence (hence also for Measure Equivalence and W*-equivalence).…

### Property (T) for uniformly bounded representations and weak*-continuity of invariant means

- Mathematics
- 2022

. For every c ≥ 1, we deﬁne a strengthening of Kazhdan’s Property (T) by con-sidering uniformly bounded representations π with ﬁxed bound | π | ≤ c . We characterise this property by the existence of…

### First $\ell^2$-Betti numbers and proper proximality

- Mathematics
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. We show that for a countable exact group, having positive ﬁrst ℓ 2 -Betti number implies proper proximality in this sense of [BIP21]. This is achieved by showing a cocycle super-rigidty result for…

### Weak amenability of free products of hyperbolic and amenable groups

- MathematicsGlasgow Mathematical Journal
- 2022

Abstract We show that if G is an amenable group and H is a hyperbolic group, then the free product
$G\ast H$
is weakly amenable. A key ingredient in the proof is the fact that
$G\ast H$
is orbit…

### Proper proximality in non-positive curvature

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- 2020

Proper proximality of a countable group is a notion that was introduced by Boutonnet, Ioana and Peterson as a tool to study rigidity properties of certain von Neumann algebras associated to groups or…

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