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Adrian Ioana

Publications and source records attributed to Adrian Ioana.

At least 19 recordsLinked to original sources
arXiv2026

A class of II$_1$ factors without non-trivial crossed product decompositions

We introduce a class of separable II$_1$ factors $M$ admitting no non-trivial crossed product decompositions: $M\not\cong B\rtimes_σG$, for any trace preserving action $G\curvearrowright^σ(B,τ)$ of an infinite countable group $G$ on a tracial von Neumann algebra $(B,τ)$. These provide the first examples of II$_1$ factors that do not arise as crossed products of noncommutative dynamical systems. Our approach relies on a novel construction of separable II$_1$ factors $M$ whose embeddings into their tensor product square $M\overline{\otimes}M$ all arise from the canonical embeddings $x\mapsto x\otimes 1$ and $x\mapsto 1\otimes x$.

math.OA↗
arXiv2026

A continuum of non-measure equivalent groups

We construct a continuum sized family $\{G_x\}_{x\in\{0,1\}^{\mathbb N}}$ of pairwise non-measure equivalent countable groups which have property (T) (hence are finitely generated), have zero $\ell^2$-Betti numbers of all orders, and are torsion-free. We also prove that the equivalence relation $\simeq^{\mathrm{fg}}_{\mathrm{ME}}$ of measure equivalence between finitely generated groups is non-smooth, resolving a question of S. Thomas. Our proof moreover shows that $\simeq^{\mathrm{fg}}_{\mathrm{ME}}$ sits above every countable Borel equivalence relation in the Borel reducibility hierarchy.

math.GR↗
arXiv2026

Amenable absorption in von Neumann algebras of hyperbolic groups

We prove that the von Neumann algebra $\cL(G)$ associated with any hyperbolic group $G$ satisfies the following \emph{amenable absorption property}: for any infinite maximal amenable subgroup $H \leqslant G$ and any amenable von Neumann subalgebra $\mathcal{Q} \subset \cL(G)$ with diffuse intersection with $\cL(H)$, one must have $\mathcal{Q} \subset \cL(H)$. This strengthens a result of Boutonnet and Carderi \cite{BC2}. We also establish similar amenable absorption results for the broader class of acylindrically hyperbolic groups, including relatively hyperbolic groups, mapping class groups, and limit groups.

math.OA↗
arXiv2026

Weyl groups and rigidity of von Neumann algebras

Let $G$ be a noncompact semisimple algebraic group with trivial center, $S < G$ a maximal split torus, $H < G$ the centralizer of $S$ in $G$ and $Γ< G$ an irreducible lattice. Consider the group measure space von Neumann algebra $\mathscr M = \operatorname{L}(Γ\curvearrowright G/H)$ associated with the nonsingular action $Γ\curvearrowright G/H$ and regard the group von Neumann algebra $M = \operatorname{L}(Γ)$ as a von Neumann subalgebra $M \subset \mathscr M$. We show that the group $\operatorname{Aut}_M(\mathscr M)$ of all unital normal $\ast$-automorphisms of $\mathscr M$ acting identically on $M$ is isomorphic to the Weyl group $\mathscr W_G$ of the semisimple algebraic group $G$. Our main theorem is a noncommutative analogue of a rigidity result of Bader-Furman-Gorodnik-Weiss for group actions on algebraic homogeneous spaces and moreover gives new insight towards Connes' rigidity conjecture for higher rank lattices.

math.OA↗
arXiv2026

Characters of surface groups

We initiate the study of characters of surface groups and their corresponding tracial representations. We show that any tracial representation can be approximated arbitrarily well in the Wasserstein topology by factorial tracial representations with spectral gap. In particular, we deduce that the space of traces of a surface group is the Poulsen simplex, thereby resolving positively a question posed by Orovitz, Slutsky, and the third author.

math.GR↗
arXiv2025

Wreath-like products of groups and their von Neumann algebras III: Embeddings

For a class of wreath-like product groups with property (T), we describe explicitly all the embeddings between their von Neumann algebras. This allows us to provide a continuum of ICC groups with property (T) whose von Neumann algebras are pairwise non (stably) embeddable. We also give a construction of groups in this class only having inner injective homomorphisms. As an application, we obtain examples of group von Neumann algebras which admit only inner endomorphisms.

math.OA↗
arXiv2025

Rigidity for graph product von Neumann algebras

We establish rigidity theorems for graph product von Neumann algebras $M_Γ=*_{v,Γ}M_v$ associated to finite simple graphs $Γ$ and families of tracial von Neumann algebras $(M_v)_{v\inΓ}$. We consider the following three broad classes of vertex algebras: diffuse, diffuse amenable, and II$_1$ factors. In each of these three regimes, we exhibit a large class of graphs $Γ,Λ$ for which the following holds: any isomorphism $θ$ between $M_Γ$ and $N_Λ$ ensures the existence of a graph isomorphism $α:Γ\toΛ$, and tight relations between $θ(M_v)$ and $N_{α(v)}$ for every vertex $v\inΓ$, ranging from strong intertwining in both directions (in the sense of Popa), to unitary conjugacy in some cases. Our results lead to a wide range of applications to the classification of graph product von Neumann algebras and the calculation of their symmetry groups. First, we obtain general classification theorems for von Neumann algebras of right-angled Artin groups and of graph products of ICC groups. We also provide a new family of II$_1$ factors with trivial fundamental group, including all graph products of II$_1$ factors over graphs with girth at least $5$ and no vertices of degree $0$ or $1$. Finally, we compute the outer automorphism group of certain graph products of II$_1$ factors.

math.OA↗
arXiv2024

Trace spaces of full free product $C^*$-algebras

We study the space of traces associated with arbitrary full free products of unital, separable $C^*$-algebras. We show that, unless certain basic obstructions (which we fully characterize) occur, the space of traces always results in the same object: the Poulsen simplex, that is, the unique infinite-dimensional metrizable Choquet simplex whose extreme points are dense. Moreover, we show that whenever such a trace space is the Poulsen simplex, the extreme points are dense in the Wasserstein topology. Concretely for the case of groups, we find that, unless the trivial character is isolated in the space of characters, the space of traces of any free product of non-trivial countable groups is the Poulsen simplex. Our main technical contribution is a new perturbation result for pairs of von Neumann subalgebras $(M_{1},M_{2})$ of a tracial von Neumann algebra $M$, providing necessary conditions under which $M_{1}$ and a small unitary perturbation of $M_{2}$ generate a II$_{1}$ factor.

math.OA↗
arXiv2023

Asymptotic freeness in tracial ultraproducts

We prove novel asymptotic freeness results in tracial ultraproduct von Neumann algebras. In particular, we show that whenever $M = M_1 \ast M_2$ is a tracial free product von Neumann algebra and $u_1 \in \mathscr U(M_1)$, $u_2 \in \mathscr U(M_2)$ are Haar unitaries, the relative commutants $\{u_1\}' \cap M^{\mathcal U}$ and $\{u_2\}' \cap M^{\mathcal U}$ are freely independent in the ultraproduct $M^{\mathcal U}$. Our proof relies on Mei-Ricard's results [MR16] regarding $\operatorname{L}^p$-boundedness (for all $1 < p < +\infty$) of certain Fourier multipliers in tracial amalgamated free products von Neumann algebras. We derive two applications. Firstly, we obtain a general absorption result in tracial amalgamated free products that recovers several previous maximal amenability/Gamma absorption results. Secondly, we prove a new lifting theorem which we combine with our asymptotic freeness results and Chifan-Ioana-Kunnawalkam Elayavalli's recent construction [CIKE22] to provide the first example of a ${\rm II_1}$ factor that does not have property Gamma and is not elementary equivalent to any free product of diffuse tracial von Neumann algebras.

math.OA↗
arXiv2023

Existential closedeness and the structure of bimodules of II$_1$ factors

We prove that if a separable II$_1$ factor $M$ is existentially closed, then every $M$-bimodule is weakly contained in the trivial $M$-bimodule, $\text{L}^2(M)$, and, equivalently, every normal completely positive map on $M$ is a pointwise 2-norm limit of maps of the form $x\mapsto\sum_{i=1}^ka_i^*xa_i$, for some $k\in\mathbb N$ and $(a_i)_{i=1}^k\subset M$. This provides the first examples of non-hyperfinite separable II$_1$ factors $M$ with the latter properties. We also obtain new characterizations of $M$-bimodules which are weakly contained in the trivial or coarse $M$-bimodule and of relative amenability inside $M$. Additionally, we give an operator algebraic presentation of the proof of the existence of existentially closed II$_1$ factors. While existentially closed II$_1$ factors have property Gamma, by adapting this proof we construct non-Gamma II$_1$ factors which are existentially closed in every weakly coarse extension.

math.OA↗
arXiv2023

Wreath-like products of groups and their von Neumann algebras I: $W^\ast$-superrigidity

We introduce a new class of groups called wreath-like products. These groups are close relatives of the classical wreath products and arise naturally in the context of group theoretic Dehn filling. Unlike ordinary wreath products, many wreath-like products have Kazhdan's property (T). In this paper, we prove that any group $G$ in a natural family of wreath-like products with property (T) is W$^*$-superrigid: the group von Neumann algebra $\text{L}(G)$ remembers the isomorphism class of $G$. This allows us to provide the first examples (in fact, $2^{\aleph_0}$ pairwise non-isomorphic examples) of W$^*$-superrigid groups with property (T).

math.OA↗
arXiv2023

An exotic II$_1$ factor without property Gamma

We introduce a new iterative amalgamated free product construction of II$_1$ factors, and use it to construct a separable II$_1$ factor which does not have property Gamma and is not elementarily equivalent to the free group factor $\text{L}(\mathbb F_n)$, for any $2\leq n\leq \infty$. This provides the first explicit example of two non-elementarily equivalent II$_1$ factors without property Gamma. Moreover, our construction also provides the first explicit example of a II$_1$ factor without property Gamma that is also not elementarily equivalent to any ultraproduct of matrix algebras. Our proofs use a blend of techniques from Voiculescu's free entropy theory and Popa's deformation/rigidity theory.

math.OA↗
arXiv2022

Orbit equivalence rigidity of irreducible actions of right-angled Artin groups

Let $G_Γ\curvearrowright X$ and $G_Λ\curvearrowright Y$ be two free measure-preserving actions of one-ended right-angled Artin groups with trivial center on standard probability spaces. Assume they are irreducible, i.e. every element from a standard generating set acts ergodically. We prove that if the two actions are stably orbit equivalent (or merely stably $W^*$-equivalent), then they are automatically conjugate through a group isomorphism between $G_Γ$ and $G_Λ$. Through work of Monod and Shalom, we derive a superrigidity statement: if the action $G_Γ\curvearrowright X$ is stably orbit equivalent (or merely stably $W^*$-equivalent) to a free, measure-preserving, mildly mixing action of a countable group, then the two actions are virtually conjugate. We also use works of Popa and Ioana-Popa-Vaes to establish the $W^*$-superrigidity of Bernoulli actions of all ICC groups having a finite generating set made of infinite-order elements where two consecutive elements commute, and one has a nonamenable centralizer: these include one-ended non-abelian right-angled Artin groups, but also many other Artin groups and most mapping class groups of finite-type surfaces.

math.GR↗
arXiv2022

Tensor product indecomposability results for existentially closed factors

In the first part of the paper we survey several results from Popa's deformation/rigidity theory on the classification of tensor product decompositions of large natural classes of II$_1$ factors. Using a mélange of techniques from deformation/rigidity theory, model theory, and the recent works \cite{CIOS21,CDI22} we highlight an uncountable family of existentially closed II$_1$ factors $M$ which do not admit tensor product decompositions $M= P\bar \otimes Q$ into diffuse factors where $Q$ is full. In the last section we discuss several open problems regarding the structural theory of existentially closed factors.

math.OA↗
arXiv2022

Embedding universality for II$_1$ factors with property (T)

We prove that every separable tracial von Neumann algebra embeds into a II$_1$ factor with property (T) which can be taken to have trivial outer automorphism and fundamental groups. We also establish an analogous result for the trivial extension over a non-atomic probability space of every countable p.m.p. equivalence relation. These results are obtained by using the class of wreath-like product groups introduced recently in \cite{CIOS21}.

math.OA↗
arXiv2021

Almost commuting matrices and stability for product groups

We prove that any product of two non-abelian free groups, $Γ=\mathbb F_m\times\mathbb F_k$, for $m,k\geq 2$, is not Hilbert-Schmidt stable. This means that there exist asymptotic representations $π_n:Γ\rightarrow \text{U}({d_n})$ with respect to the normalized Hilbert-Schmidt norm which are not close to actual representations. As a consequence, we prove the existence of contraction matrices $A,B$ such that $A$ almost commutes with $B$ and $B^*$, with respect to the normalized Hilbert-Schmidt norm, but $A,B$ are not close to any matrices $A',B'$ such that $A'$ commutes with $B'$ and $B'^*$. This settles in the negative a natural version of a question concerning almost commuting matrices posed by Rosenthal in 1969.

math.OA↗
arXiv2021

Cocycle superrigidity for profinite actions of irreducible lattices

Let $Γ$ be an irreducible lattice in a product of two locally compact groups and assume that $Γ$ is densely embedded in a profinite group $K$. We give necessary conditions which imply that the left translation action $Γ\curvearrowright K$ is "virtually" cocycle superrigid: any cocycle $w:Γ\times K\rightarrowΔ$ with values in a countable group $Δ$ is cohomologous to a cocycle which factors through the map $Γ\times K\rightarrowΓ\times K_0$, for some finite quotient group $K_0$ of $K$. As a corollary, we deduce that any ergodic profinite action of $Γ=\text{SL}_2(\mathbb Z[S^{-1}])$ is virtually cocycle superrigid and virtually W$^*$-superrigid, for any finite nonempty set of primes $S$.

math.DS↗