Searcharxiv⌕ Search

arXiv subjects

Andrea Vaccaro

Publications and source records attributed to Andrea Vaccaro.

At least 19 recordsLinked to original sources
arXiv2026

Stable rank one, tracial local homogeneity and uniform property $Γ$

We prove that separable, simple, unital, non-elementary, stably finite C*-algebras that have stable rank one, and that have locally finite nuclear dimension in a tracial sense, have uniform property $Γ$. In particular, Villadsen algebras of the first type and crossed products of free minimal actions of FC (in particular, abelian) groups on compact metric spaces have uniform property $Γ$. This implies that all these C*-algebras satisfy the Toms-Winter conjecture, a fact already known for C*-algebras with stable rank one and locally finite nuclear dimension, and here recovered via a different approach.

math.OA↗
arXiv2026

Hyper-u-amenablity and Hyperfiniteness of Treeable Equivalence Relations

We introduce the notions of u-amenability and hyper-u-amenability for countable Borel equivalence relations, strong forms of amenability that are implied by hyperfiniteness. We show that treeable, hyper-u-amenable countable Borel equivalence relations are hyperfinite. One of the corollaries that we get is that if a countable Borel equivalence relation is measure-hyperfinite and equal to the orbit equivalence relation of a free continuous action of a virtually free group on a $σ$-compact Polish space, then it is hyperfinite. We also obtain that if a countable Borel equivalence relation is treeable and equal to the orbit equivalence relation of a Borel action of an amenable group on a standard Borel space, or if it is treeable, amenable and Borel bounded, then it is hyperfinite.

math.LO↗
arXiv2025

Continuous Selection of Unitaries in II$_1$ Factors

We prove continuous-valued analogues of the basic fact that Murray-von Neumann subequivalence of projections in II$_1$ factors is completely determined by tracial evaluations. We moreover use this result to solve the so-called trace problem in the case of factorial trivial $W^\ast$-bundles whose base space has covering dimension at most 1. Our arguments are based on applications of a continuous selection theorem due to Michael to von Neumann algebras.

math.OA↗
arXiv2025

Probably isomorphic structures

Two structures $M, N$ in the same language are called probably isomorphic if they (or, in case of metric structures, their completions) are isomorphic after forcing with the Lebesgue measure algebra. We show that, if $M$ and $N$ are discrete structures, or extremal models of a non-degenerate simplicial theory, then $M$ and $N$ are probably isomorphic if and only if $L^1([0,1], M) \cong L^1([0,1], N)$. We moreover employ some of the set-theoretic arguments used to prove the aforementioned result to characterize when nontrivial ultraproducts of diffuse von Neumann algebras are tensorially prime.

math.LO↗
arXiv2025

Corona Rigidity

We give a unified overview of the study of the effects of additional set theoretic axioms on quotient structures. Our focus is on rigidity, measured in terms of existence (or rather non-existence) of suitably non-trivial automorphisms of the quotients in question. A textbook example for the study of this topic is the Boolean algebra $\mathcal{P}(\mathbb{N})/\text{Fin}$, whose behavior is the template around which this survey revolves: Forcing axioms imply that all of its automorphisms are trivial, in the sense that they are induced by almost permutations of $\mathbb{N}$, while under the Continuum Hypothesis this rigidity fails and $\mathcal{P}(\mathbb{N})/\text{Fin}$ admits uncountably many non-trivial automorphisms. We consider far-reaching generalisations of this phenomenon and present a wide variety of situations where analogous patterns persist, focusing mainly (but not exclusively) on the categories of Boolean algebras, Čech-Stone remainders, and $\mathrm{C}^\ast$-algebras. We survey the state of the art and the future prospects of this field, discussing the major open problems and outlining the main ideas of the proofs whenever possible.

math.LO↗
arXiv2025

Uniform property $Γ$ and the small boundary property

We prove that, for a free action $α\colon G \curvearrowright X$ of a countably infinite discrete amenable group on a compact metric space, the small boundary property is implied by uniform property $Γ$ of the Cartan subalgebra $(C(X) \subseteq C(X) \rtimes_αG)$. The reverse implication has been demonstrated by Kerr and Szabó for free actions, from which we obtain that these two conditions are equivalent. We moreover show that, if $α$ is also minimal, then almost finiteness of $α$ is implied by tracial $\mathcal{Z}$-stability of the subalgebra $(C(X) \subseteq C(X) \rtimes_αG)$. The reverse implication is due to Kerr, resulting in the equivalence of these two properties as well. As an application, we prove that if $α\colon G \curvearrowright X$ and $β\colon H \curvearrowright Y$ are free actions and $α$ has the small boundary property, then $α\times β\colon G \times H \curvearrowright X \times Y$ has the small boundary property. An analogous permanence property is obtained for almost finiteness in case $α$ and $β$ are free minimal actions.

math.OA↗
arXiv2025

Surfaces and other Peano Continua with no Generic Chains

The space of chains on a compact connected space encodes all the different ways of continuously growing out of a point until exhausting the space. A chain is \emph{generic} if its orbit under the action of the underlying homeomorphism group is comeager. In this paper we show that a large family of topological spaces do not have a generic chain: in addition to all manifolds of dimension at least 3, for which the result was already known, our theorem covers all compact surfaces except for the sphere and the real projective plane - for which the question remains open - as well as all other homogeneous Peano continua, circle excluded. If the spaces are moreover strongly locally homogeneous, which is the case for any closed manifold and the Menger curve, we prove that chains cannot be classified up to homeomorphism by countable structures, and that the underlying homeomorphism groups have non-metrizable universal minimal flows, with all orbits meager, in contrast to the case of 1-dimensional manifolds. The proof of the main result is of combinatorial nature, and it relies on the creation of a dictionary between open sets of chains on one side, and walks on finite connected graphs on the other.

math.DS↗
arXiv2024

Actions on classifiable C*-algebras without equivariant property (SI)

We exhibit examples of actions of countable discrete groups on both simple and non-simple nuclear stably finite C*-algebras that are tracially amenable but not amenable. We furthermore obtain that, under the additional assumption of strict comparison, amenability is equivalent to tracial amenability plus the equivariant analogue of Matui--Sato's property (SI). By virtue of this equivalence, our construction yields the first known examples of actions on classifiable C*-algebras that do not have equivariant property (SI). We moreover show that such actions can be chosen to absorb the trivial action on the universal UHF algebra, thus showing that equivariant $\mathcal{Z}$-stability does not in general imply equivariant property (SI).

math.OA↗
arXiv2024

Tracially amenable actions and purely infinite crossed products

We introduce the notion of tracial amenability for actions of discrete groups on unital, tracial C$^*$-algebras, as a weakening of amenability where all the relevant approximations are done in the uniform trace norm. We characterize tracial amenability with various equivalent conditions, including topological amenability of the induced action on the trace space. Our main result concerns the structure of crossed products: for groups containing the free group $F_2$, we show that outer, tracially amenable actions on simple, unital, $\mathcal{Z}$-stable C$^*$-algebras always have purely infinite crossed products. Finally, we give concrete examples of tracially amenable actions of free groups on simple, unital AF-algebras.

math.OA↗
arXiv2023

Ultraproducts of factorial $W^*$-bundles

This paper investigates factorial $W^*$-bundles and their ultraproducts. More precisely, a $W^*$-bundle is factorial if the von Neumann algebras associated to its fibers are all factors. Let $M$ be the tracial ultraproduct of a family of factorial $W^*$-bundles over compact Hausdorff spaces with finite, uniformly bounded covering dimensions. We prove that in this case the set of limit traces in $M$ is weak$^*$-dense in the trace space $T(M)$. This in particular entails that $M$ is factorial. We also provide, on the other hand, an example of ultraproduct of factorial $W^*$-bundles which is not factorial. Finally, we obtain some results of model-theoretic nature: if $A$ and $B$ are exact, $\mathcal{Z}$-stable $C^*$-algebras, or if they both have strict comparison, then $A \equiv B$ implies that $T(A)$ is Bauer if and only if $T(B)$ is. If moreover both $T(A)$ and $T(B)$ are Bauer simplices and second countable, then the sets of extreme traces $\partial_e T(A)$ and $\partial_e T(B)$ have the same covering dimension.

math.OA↗
arXiv2022

Dynamical comparison and $\mathcal{Z}$-stability for crossed products of simple $C^*$-algebras

We establish $\mathcal{Z}$-stability for crossed products of outer actions of amenable groups on $\mathcal{Z}$-stable $C^*$-algebras under a mild technical assumption which we call McDuff property with respect to invariant traces. We obtain such result using a weak form of dynamical comparison, which we verify in great generality. We complement our results by proving that McDuffness with respect to invariant traces is automatic in many cases of interest. This is the case, for instance, for every action of an amenable group $G$ on a classifiable $C^*$-algebra $A$ whose trace space $T(A)$ is a Bauer simplex with finite dimensional boundary $\partial_e T(A)$, and such that the induced action $G\curvearrowright \partial_eT(A)$ is free. If $G = \mathbb{Z}^d$ and the action $G\curvearrowright \partial_eT(A)$ is free and minimal, then we obtain McDuffness with respect to invariant traces, and thus $\mathcal{Z}$-stability of the corresponding crossed product, also in case $\partial_e T(A)$ has infinite covering dimension.

math.OA↗
arXiv2022

Games on AF-algebras

We analyze $\mathrm{C}^\ast$-algebras, particularly AF-algebras, and their $K_0$-groups in the context of the infinitary logic $\mathcal{L}_{ω_1 ω}$. Given two separable unital AF-algebras $A$ and $B$, and considering their $K_0$-groups as ordered unital groups, we prove that $K_0(A) \equiv_{ω\cdot α} K_0(B)$ implies $A \equiv_αB$, where $M \equiv_βN$ means that $M$ and $N$ agree on all sentences of quantifier rank at most $β$. This implication is proved using techniques from Elliott's classification of separable AF-algebras, together with an adaptation of the Ehrenfeucht-Fraïssé game to the metric setting. We use moreover this result to build a family $\{ A_α\}_{α< ω_1}$ of pairwise non-isomorphic separable simple unital AF-algebras which satisfy $A_α\equiv_αA_β$ for every $α< β$. In particular, we obtain a set of separable simple unital AF-algebras of arbitrarily high Scott rank. Next, we give a partial converse to the aforementioned implication, showing that $A \otimes \mathcal{K} \equiv_{ω+ 2 \cdot α+2} B \otimes \mathcal{K}$ implies $K_0(A) \equiv_αK_0(B)$, for every unital $\mathrm{C}^\ast$-algebras $A$ and $B$.

math.LO↗
arXiv2022

Extending depolarized DLS measurements to turbid samples

The application of dynamic light scattering to soft matter systems has strongly profited from advanced approaches such as the so-called modulated 3D cross correlation technique (mod3D-DLS) that suppress contributions from multiple scattering, and can therefore be used for the characterization of turbid samples. Here we now extend the possibilities of this technique to allow for depolarized light scattering (Mod3D-DDLS) and thus obtain information on both translational and rotational diffusion, which is important for the characterization of anisotropic particles. We describe the required optical design and test the performance of the approach for increasingly turbid samples using well defined anisotropic colloidal models systems. Our measurements demonstrate that 3D-DDLS experiments can be performed successfully for samples with a reduced transmission due to multiple scattering as low as 1\%. We compare the results from this approach with those obtained by standard DDLS experiments, and point out the importance of using an appropriate optical design when performing depolarized dynamic light scattering experiments with turbid systems.

cond-mat.soft↗
arXiv2021

Strongly outer actions of amenable groups on $\mathcal{Z}$-stable nuclear $C^*$-algebras

Let $A$ be a separable, unital, simple, $\mathcal{Z}$-stable, nuclear $C^*$-algebra, and let $α\colon G\to \mathrm{Aut}(A)$ be an action of a discrete, countable, amenable group. Suppose that the orbits of the action of $G$ on $T(A)$ are finite and that their cardinality is bounded. We show that $α$ is strongly outer if and only if $α\otimes\mathrm{id}_{\mathcal{Z}}$ has the weak tracial Rokhlin property. If $G$ is moreover residually finite, these conditions are also equivalent to $α\otimes\mathrm{id}_{\mathcal{Z}}$ having finite Rokhlin dimension (in fact, at most 2). If $\partial_eT(A)$ is furthermore compact, has finite covering dimension, and the orbit space $\partial_eT(A)/G$ is Hausdorff, we generalize results by Matui and Sato to show that $α$ is cocycle conjugate to $α\otimes\mathrm{id}_{\mathcal{Z}}$, even if $α$ is not strongly outer. In particular, in this case the equivalences above hold for $α$ in place of $α\otimes\mathrm{id}_{\mathcal{Z}}$. In the course of the proof, we develop equivariant versions of complemented partitions of unity and uniform property $Γ$ as technical tools of independent interest.

math.OA↗
arXiv2021

Trivial Endomorphisms of the Calkin Algebra

We prove that it is consistent with ZFC that every unital endomorphism of the Calkin algebra $\mathcal{Q}(H)$ is unitarily equivalent to an endomorphism of $\mathcal{Q}(H)$ which is liftable to a unital endomorphism of $\mathcal{B}(H)$. We use this result to classify all unital endomorphisms of $\mathcal{Q}(H)$ up to unitary equivalence by the Fredholm index of the image of the unilateral shift. As a further application, we show that it is consistent with ZFC that the class of $\mathrm{C}^\ast$-algebras that embed into $\mathcal{Q}(H)$ is not closed under tensor product nor countable inductive limit.

math.OA↗
arXiv2020

Trace spaces of counterexamples to Naimark's Problem

A counterexample to Naimark's problem is a $C^\ast$-algebra that is not isomorphic to the algebra of compact operators on some Hilbert space, yet still has only one irreducible representation up to unitary equivalence. It is well-known that such algebras must be nonseparable, and in 2004 Akemann and Weaver used the diamond principle (a set theoretic axiom independent from ZFC) to give the first counterexamples. For any such counterexample $A$, the unitary group $U(A)$ acts transitively on the pure states, which are the extreme points of the state space $S(A)$. It is conceivable that this implies (as happens for finite-dimensional simplexes) that the action of $U(A)$ on $S(A)$ has at most one fixed point, i.e. $A$ has at most one trace. We give a strong negative answer here assuming diamond. In particular, we adapt the Akemann-Weaver construction to show that the trace space of a counterexample to Naimark's problem can be affinely homeomorphic to any metrizable Choquet simplex, and can also be nonseparable.

math.OA↗
arXiv2019

Obstructions to lifting abelian subalgebras of corona algebras

Let $A$ be a non-commutative, non-unital $\mathrm{C}^\ast$-algebra. Given a set of commuting positive elements in the corona algebra $Q(A)$, we study some obstructions to the existence of a commutative lifting of such set to the multiplier algebra $M(A)$. Our focus are the obstructions caused by the size of the collection we want to lift. It is known that no obstacles show up when lifting a countable family of commuting projections, or of pairwise orthogonal positive elements. However, this is not the case for larger collections. We prove in fact that for every primitive, non-unital, $σ$-unital $\mathrm{C}^\ast$-algebra $A$, there exists an uncountable set of pairwise orthogonal positive elements in $Q(A)$ such that no uncountable subset of it can be lifted to a set of commuting elements of $M(A)$. Moreover, the positive elements in $Q(A)$ can be chosen to be projections if $A$ has real rank zero.

math.OA↗