arXiv subjects

Stefaan Vaes

Publications and source records attributed to Stefaan Vaes.

At least 19 recordsLinked to original sources

arXiv2025

W*-correlations of II$_1$ factors and rigidity of tensor products and graph products

A variant of Gromov's notion of measure equivalence for groups has been introduced for II$_1$ factors under different names. We propose the terminology of W*-correlated II$_1$ factors. We prove rigidity results up to W*-correlations for tensor products and graph products of II$_1$ factors. As a consequence, we construct the first uncountable family of discrete groups $\Gamma$ that are not von Neumann equivalent, which means that their group von Neumann algebras $L(\Gamma)$ are not W*-correlated, and which implies that these groups are neither measure equivalent, nor have isomorphic or virtually isomorphic group von Neumann algebras.

math.OA↗

arXiv2025

W*-superrigidity for discrete quantum groups

A discrete group $G$ is called W*-superrigid if the group $G$ can be entirely recovered from the ambient group von Neumann algebra $L(G)$. We introduce an analogous notion for discrete quantum groups. We prove that this strengthened quantum W*-superrigidity property holds for a natural family of co-induced discrete quantum groups. We also prove that, remarkably, most existing families of W*-superrigid groups are not quantum W*-superrigid.

math.OA↗

arXiv2024

Factoriality of twisted locally compact group von Neumann algebras

In this short note, we construct an exotic example of a locally compact group $G$ with a Borel $2$-cocycle $\omega$ such that the non-twisted group von Neumann algebra $L(G)$ is a factor, while the twisted group von Neumann algebra $L_\omega(G)$ has a diffuse center.

math.OA↗

arXiv2024

W*-superrigidity for groups with infinite center

We construct discrete groups $G$ with infinite center that are nevertheless W*-superrigid, meaning that the group von Neumann algebra $L(G)$ fully remembers the group $G$. We obtain these rigidity results both up to isomorphisms and up to virtual isomorphisms of the groups and their von Neumann algebras. Our methods combine rigidity results for the quotient of these groups by their center with rigidity results for their 2-cohomology.

math.OA↗

arXiv2024

Borel fields and measured fields of Polish spaces, Banach spaces, von Neumann algebras and C*-algebras

Several recent articles in operator algebras make a nontrivial use of the theory of measurable fields of von Neumann algebras $(M_x)_{x \in X}$ and related structures. This includes the associated field $(\text{Aut}\ M_x)_{x \in X}$ of automorphism groups and more general measurable fields of Polish groups with actions on Polish spaces. Nevertheless, a fully rigorous and at the same time sufficiently broad and flexible theory of such Borel fields and measurable fields is not available in the literature. We fill this gap in this paper and include a few counterexamples to illustrate the subtlety: for instance, for a Borel field $(M_x)_{x \in X}$ of von Neumann algebras, the field of Polish groups $(\text{Aut}\ M_x)_{x \in X}$ need not be Borel.

math.OA↗

arXiv2024

Every locally compact group is the outer automorphism group of a II$_1$ factor

We prove that every locally compact second countable group $G$ arises as the outer automorphism group Out $M$ of a II$_1$ factor, which was so far only known for totally disconnected groups, compact groups and a few isolated examples. We obtain this result by proving that every locally compact second countable group is a centralizer group, a class of Polish groups that arise naturally in ergodic theory and that may all be realized as Out $M$.

math.GR↗

arXiv2024

W*-superrigidity for cocycle twisted group von Neumann algebras

We construct countable groups $G$ with the following new degree of W*-superrigidity: if $L(G)$ is virtually isomorphic, in the sense of admitting a bifinite bimodule, with any other group von Neumann algebra $L(\Lambda)$, then the groups $G$ and $\Lambda$ must be virtually isomorphic. Moreover, we allow both group von Neumann algebras to be twisted by an arbitrary $2$-cocycle. We also give examples of II$_1$ factors $N$ that are indecomposable in every sense: they are not virtually isomorphic to any cocycle twisted groupoid von Neumann algebra.

math.OA↗

arXiv2023

Ergodic states on type III$_1$ factors and ergodic actions

Since the early days of Tomita-Takesaki theory, it is known that a von Neumann algebra $M$ that admits a state $\varphi$ with trivial centralizer $M_\varphi$ must be a type III$_1$ factor, but the converse remained open. We solve this problem and prove that such ergodic states form a dense $G_\delta$ set among all faithful normal states on any III$_1$ factor with separable predual. Through Connes' Radon-Nikodym cocycle theorem, this problem is related to the existence of ergodic cocycle perturbations for outer group actions, which we consider in the second part of the paper.

math.OA↗

arXiv2022

Measure equivalence embeddings of free groups and free group factors

We give a simple and explicit proof that the free group $\mathbb{F}_2$ admits a measure equivalence embedding into any nonamenable locally compact second countable (lcsc) group $G$. We use this to prove that every nonamenable lcsc group $G$ admits strongly ergodic actions of any possible Krieger type and admits nonamenable, weakly mixing actions with any prescribed flow of weights. We also introduce concepts of measure equivalence and measure equivalence embeddings for $II_1$ factors. We prove that a $II_1$ factor $M$ is nonamenable if and only if the free group factor $L(\mathbb{F}_2)$ admits a measure equivalence embedding into $M$. We prove stability of property (T) and the Haagerup property under measure equivalence of $II_1$ factors.

math.GR↗

arXiv2022

Quantum automorphism groups of connected locally finite graphs and quantizations of discrete groups

We construct for every connected locally finite graph $\Pi$ the quantum automorphism group $\text{QAut}\ \Pi$ as a locally compact quantum group. When $\Pi$ is vertex transitive, we associate to $\Pi$ a new unitary tensor category $\mathcal{C}(\Pi)$ and this is our main tool to construct the Haar functionals on $\text{QAut}\ \Pi$. When $\Pi$ is the Cayley graph of a finitely generated group, this unitary tensor category is the representation category of a compact quantum group whose discrete dual can be viewed as a canonical quantization of the underlying discrete group. We introduce several equivalent definitions of quantum isomorphism of connected locally finite graphs $\Pi$, $\Pi'$ and prove that this implies monoidal equivalence of $\text{QAut}\ \Pi$ and $\text{QAut}\ \Pi'$.

math.QA↗

arXiv2022

W*-rigidity paradigms for embeddings of II$_1$ factors

We undertake a systematic study of W*-rigidity paradigms for the embeddability relation $\hookrightarrow$ between separable II$_1$ factors and its stable version $\hookrightarrow_s$, obtaining large families of non stably isomorphic II$_1$ factors that are mutually embeddable and families of II$_1$ factors that are mutually non stably embeddable. We provide an augmentation functor $G \mapsto H_G$ from the category of groups into icc groups, so that $L(H_{G_1}) \hookrightarrow_s L(H_{G_2})$ iff $G_1 \hookrightarrow G_2$. We construct complete intervals of II$_1$ factors, including a strict chain of II$_1$ factors $(M_k)_{k\in \mathbb{Z}}$ with the property that if $N$ is any II$_1$ factor with $M_i \hookrightarrow_s N$ and $N \hookrightarrow_s M_{j}$, then $N \cong M_k^t$ for some $i \leq k \leq j$ and $t > 0$.

math.OA↗

arXiv2022

Orbit equivalence superrigidity for type III$_0$ actions

We prove the first orbit equivalence superrigidity results for actions of type III$_\lambda$ when $\lambda \neq 1$. These actions arise as skew products of actions of dense subgroups of $SL(n,\mathbb{R})$ on the sphere $S^{n-1}$ and they can have any prescribed associated flow.

math.OA↗

arXiv2022

Classification results for nonsingular Bernoulli crossed products

We prove rigidity and classification results for type III factors given by nonsingular Bernoulli actions of the free groups and more general free product groups. This includes a large family of nonisomorphic Bernoulli crossed products of type III$_1$ that cannot be distinguished by Connes $\tau$-invariant. These are the first such classification results beyond the well studied probability measure preserving case.

math.OA↗

arXiv2022

Spectral gap and strict outerness for actions of locally compact groups on full factors

We prove that an outer action of a locally compact group $G$ on a full factor $M$ is automatically strictly outer, meaning that the relative commutant of $M$ in the crossed product is trivial. If moreover the image of $G$ in the outer automorphism group $\operatorname{Out} M$ is closed, we prove that the crossed product remains full. We obtain this result by proving that the inclusion of $M$ in the crossed product automatically has a spectral gap property. Such results had only been proven for actions of discrete groups and for actions of compact groups, by using quite different methods in both cases. Even for the canonical Bogoljubov actions on free group factors or free Araki-Woods factors, these results are new.

math.OA↗

arXiv2021

Nonsingular Gaussian actions: beyond the mixing case

Every affine isometric action $α$ of a group $G$ on a real Hilbert space gives rise to a nonsingular action $\hatα$ of $G$ on the associated Gaussian probability space. In the recent paper [AIM19], several results on the ergodicity and Krieger type of these actions were established when the underlying orthogonal representation $π$ of $G$ is mixing. We develop new methods to prove ergodicity when $π$ is only weakly mixing. We determine the type of $\hatα$ in full generality. Using Cantor measures, we give examples of type III$_1$ ergodic Gaussian actions of $\mathbb{Z}$ whose underlying representation is non mixing, and even has a Dirichlet measure as spectral type. We also provide very general ergodicity results for Gaussian skew product actions.

math.DS↗

arXiv2021

KMS spectra for group actions on compact spaces

Given a topologically free action of a countable group $G$ on a compact metric space $X$, there is a canonical correspondence between continuous 1-cocycles for this group action and diagonal 1-parameter groups of automorphisms of the reduced crossed product C*-algebra. The KMS spectrum is defined as the set of inverse temperatures for which there exists a KMS state. We prove that the possible KMS spectra depend heavily on the nature of the acting group $G$. For groups of subexponential growth, we prove that the only possible KMS spectra are $\{0\}$, $[0,+\infty)$, $(-\infty,0]$ and $\mathbb{R}$. For certain wreath product groups, which are amenable and of exponential growth, we prove that any closed subset of $\mathbb{R}$ containing zero arises as KMS spectrum. Finally, for certain nonamenable groups including the free group with infinitely many generators, we prove that any closed subset may arise. Besides uncovering a surprising relation between geometric group theoretic properties and KMS spectra, our results provide two simple C*-algebras with the following universality property: any closed subset (containing, resp. not containing zero) arises as the KMS spectrum of a 1-parameter group of automorphisms of this C*-algebra.

math.OA↗

arXiv2021

Superrigidity for dense subgroups of Lie groups and their actions on homogeneous spaces

An essentially free group action of $\Gamma$ on $(X,\mu)$ is called W*-superrigid if the crossed product von Neumann algebra $L^\infty(X) \rtimes \Gamma$ completely remembers the group $\Gamma$ and its action on $(X,\mu)$. We prove W*-superrigidity for a class of infinite measure preserving actions, in particular for natural dense subgroups of isometries of the hyperbolic plane. The main tool is a new cocycle superrigidity theorem for dense subgroups of Lie groups acting by translation. We also provide numerous countable type $II_1$ equivalence relations that cannot be implemented by an essentially free action of a group, both of geometric nature and through a wreath product construction.

math.OA↗

arXiv2021

Bernoulli actions of type III$_0$ with prescribed associated flow

We prove that many, but not all injective factors arise as crossed products by nonsingular Bernoulli actions of the group $\mathbb{Z}$. We obtain this result by proving a completely general result on the ergodicity, type and Krieger's associated flow for Bernoulli shifts with arbitrary base spaces. We prove that the associated flow must satisfy a structural property of infinite divisibility. Conversely, we prove that all almost periodic flows, as well as many other ergodic flows, do arise as associated flow of a weakly mixing Bernoulli action of any infinite amenable group. As a byproduct, we prove that all injective factors with almost periodic flow of weights are infinite tensor products of $2 \times 2$ matrices. Finally, we construct Poisson suspension actions with prescribed associated flow for any locally compact second countable group that does not have property (T).

math.DS↗