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Adriana Fernández Quero

Publications and source records attributed to Adriana Fernández Quero.

5 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

Properly proximal countable measured groupoids

We introduce the notion of proper proximality for countable measured groupoids, extending the corresponding notion for countable groups. We prove that this groupoid property is equivalent to a strengthened form of relative proper proximality for the associated groupoid von Neumann algebra. We investigate permanence properties and show, in particular, that proper proximality is preserved under finite direct products, formation of transformation groupoids, and measure equivalence. We also identify broad classes of properly proximal groupoids, including transverse measured groupoids and free product groupoids, and prove that inner amenable groupoids are never properly proximal.

math.OA↗
arXiv2026

W$^*$-superrigidity for property (T) groups with infinite center

We propose to study a natural version of Connes' Rigidity Conjecture that involves property (T) groups with infinite center. Utilizing techniques at the intersection of von Neumann algebras and geometric group theory, we establish several cases where this conjecture holds. In particular, we provide the first example of a W$^*$-superrigid property (T) group with infinite center.

math.OA↗
arXiv2026

Non-amenable C$^*$-superrigid groups that are not W$^*$-superrigid

Using techniques at the intersection of deformation/rigidity theory, geometric group theory, and the theory of $C^*$-algebras, we construct a continuum of nonamenable groups $G$ that can be completely reconstructed from their reduced $C^*$-algebras $C_r^*(G)$, but not from their group von Neumann algebras $\mathrm{L}(G)$. These groups arise as infinite direct sums of amalgamated free product groups and constitute the first known examples of nonamenable groups exhibiting this phenomenon. In addition, we provide examples of finite direct products of amalgamated free product groups that are simultaneously $C^*$-superrigid and $W^*$-superrigid. Finally, for a fairly large subclass of these amalgamated free product groups $G$, we show that all $\ast$-endomorphisms of $C_r^*(G)$ are weakly inner.

math.OA↗
arXiv2024

Rigidity results for group von Neumann algebras with diffuse center

We introduce the first examples of groups $G$ with infinite center which in a natural sense are completely recognizable from their von Neumann algebras, $\mathcal{L}(G)$. Specifically, assume that $G=A\times W$, where $A$ is an infinite abelian group and $W$ is an ICC wreath-like product group [CIOS22a; AMCOS23] with property (T) and trivial abelianization. Then whenever $H$ is an \emph{arbitrary} group such that $\mathcal{L}(G)$ is $\ast$-isomorphic to $\mathcal L(H)$ it must be the case that $H= B \times H_0$ where $B$ is infinite abelian and $H_0$ is isomorphic to $W$. Moreover, we completely describe the $\ast$-isomorphism between $\mathcal L(G)$ and $\mathcal L(H)$. This yields new applications to the classification of group C$^*$-algebras, including examples of non-amenable groups which are recoverable from their reduced C$^*$-algebras but not from their von Neumann algebras.

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