arXiv subjects

Eduard Vilalta

Publications and source records attributed to Eduard Vilalta.

At least 19 recordsLinked to original sources

arXiv2026

Super adaptable graphs and the realization problem

We introduce and study the family of super adaptable monoids, which includes, among many others, all conical refinement monoids that are either finitely generated, or countable, primely generated, and regular. Our main result shows that any countable super adaptable monoid can be realized by a von Neumann regular ring, thus answering affirmatively the Realization problem for this class. The result is new even for the subclass of countable, primely generated, regular conical refinement monoids, which we show can be realized by both a von Neumann regular ring and a purely infinite, real rank zero C*-algebra. To prove this result, we develop a systematic framework that connects three fundamental structures: abstract monoids, separated graphs (a specific class of colored graphs), and von Neumann regular rings. By establishing functorial connections among these classes, we translate several algebraic properties of monoids into combinatorial and ring-theoretic notions. This allows us not only to prove the main realization result, but also to obtain several characterizations of super adaptable monoids. As a particular application, these can be used to provide a complete characterization of which regular monoids arise as graph monoids. To our knowledge, this work offers the first self-contained treatment in the literature where the complete realization process ---from abstract monoids through I-systems and separated graphs to regular rings and C*-algebras--- is developed sequentially and in full detail within a single text.

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arXiv2026

Z-stability of twisted group C*-algebras of nilpotent groups

We prove that the twisted group C*-algebra of a finitely generated nilpotent group is $\mathcal{Z}$-stable if and only if it is nowhere scattered, a condition that we characterize in terms of the given group and 2-cocycle. As a main application, we prove new converses to the Balian-Low Theorem for projective, square-integrable representations of nilpotent Lie groups.

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arXiv2026

Strict comparison for twisted group C*-algebras

We prove that any reduced twisted group C*-algebra of a selfless group with the rapid decay property is selfless. As an application, we show that twisted group C*-algebras of acylindrically hyperbolic groups (possibly with nontrivial finite radical) and rapid decay are pure, and hence have strict comparison.

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arXiv2026

Continuous functions over a pure C*-algebra

Let $X$ be a compact metric space, and let $A$ be a pure $\mathrm{C}^*$-algebra. We show that $C(X,A)$ is pure whenever $A$ is simple; or every quotient of $A$ is stably finite (e.g., $A$ has stable rank one). Using permanence properties of pureness, we prove that the tensor product of any such $A$ with any ASH-algebra is pure.

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arXiv2025

Centrally pure C*-algebras

We show that a separable C*-algebra $A$ is $\mathcal{Z}$-stable if and only if its uncorrected central sequence algebra $A' \cap A_{\mathcal{U}}$ is pure, if and only if Kirchberg's central sequence algebra $F(A)$ is pure. More generally, we show that a C*-algebra $A$ is separably $\mathcal{Z}$-stable if and only if the relative central sequence algebra $B' \cap A_{\mathcal{U}}$ is pure for every separable subalgebra $B \subseteq A_{\mathcal{U}}$.

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arXiv2025

An introduction to the Global Glimm Problem

The Global Glimm Problem lies at the heart of several open questions regarding regularity properties of C*-algebras. The problem has been open for over two decades, and has recently garnered significant attention due to its strong ties to non-simple versions of well-known dimension reduction phenomena for simple C*-algebras, such as the weakly purely infinite problem and the non-simple Toms-Winter conjecture. This survey offers a concise introduction to the problem, detailing some of the key concepts and methods used in the partial solutions achieved thus far. Particular emphasis is placed on the implications that a complete resolution of the Global Glimm Problem would have for other open questions in the field.

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arXiv2025

The Global Glimm Property for C*-algebras of topological dimension zero

We show that a C*-algebra with topological dimension zero has the Global Glimm Property (every hereditary subalgebra contains an almost full nilpotent element) if and only if it is nowhere scattered (no hereditary subalgebra admits a finite-dimensional representation). This solves the Global Glimm Problem in this setting. It follows that nowhere scattered C*-algebras with finite nuclear dimension and topological dimension zero are pure.

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arXiv2025

Extensions of pure C*-algebras

Given a closed ideal $I$ in a C*-algebra $A$, we show that $A$ is pure if and only if $I$ and $A/I$ are pure. More generally, we study permanence of comparison and divisibility properties when passing to extensions. As an application we show that stable multiplier algebras of reduced free group C*-algebras are pure.

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arXiv2024

Pure C*-algebras

We demonstrate that pure C*-algebras form a robust class by proving that pureness follows from very weak comparison and divisibility properties. Using this, we show that every simple, non-elementary C*-algebra with a unique quasitrace and with very mild comparison is pure, and, as a result, has strict comparison. Furthermore, sufficiently non-commutative C*-algebras of stable rank one and with weak comparison are likewise pure. We also show that adequately non-elementary C*-algebras with finite nuclear dimension are pure, which leads to the verification of the non-simple Toms-Winter conjecture for a large class of C*-algebras.

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arXiv2024

Ideals, quotients, and continuity of the Cuntz semigroup for rings

In this paper we explore which part of the ideal lattice of a general ring is parametrized by its Cuntz semigroup $\mathrm{S}(R)$ and its ambient semigroup $Λ(R)$. We identify these classes of ideals as the quasipure ideals (a generalization of pure ideals) in the case of $\mathrm{S}(R)$, and what we term decomposable ideals in the case of $Λ(R)$. For an ($s$-)unital ring $R$, the latter class exhausts all ideals of the ring. We prove that these constructions behave well with respect to quotients. In order to study the passage to inductive limits, we introduce the classes of dense and left normal rings. We show that $\mathrm{S}(R)$ is an abstract Cu-semigroup whenever $R$ is left normal and, for such rings, the assignment $R\mapsto \mathrm{S}(R)$ is continuous. We prove a parallel result for $Λ(R)$ whenever $R$ is a dense ring.

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arXiv2024

Criteria for the existence of Schwartz Gabor frames over rational lattices

We give an explicit criterion for a rational lattice in the time-frequency plane to admit a Gabor frame with window in the Schwartz class. The criterion is an inequality formulated in terms of the lattice covolume, the dimension of the underlying Euclidean space, and the index of an associated subgroup measuring the degree of non-integrality of the lattice. For arbitrary lattices we also give an upper bound on the number of windows in the Schwartz class needed for a multi-window Gabor frame.

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arXiv2024

Pure *-homomorphisms

We introduce and study a notion of pureness for *-homomorphisms and, more generally, for cpc. order-zero maps. After providing several examples of pureness, such as "$\mathcal{Z}$-stable"-like maps, we focus on the question of when pure maps factor through a pure C*-algebra. We show that, up to Cuntz equivalence, any composition of two pure maps factors through a pure object. This is used to obtain several factorization results at the level of C*-algebras.

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arXiv2023

Ranks of soft operators in nowhere scattered C*-algebras

We show that for C*-algebras with the Global Glimm Property, the rank of every operator can be realized as the rank of a soft operator, that is, an element whose hereditary sub-C*-algebra has no nonzero, unital quotients. This implies that the radius of comparison of such a C*-algebra is determined by the soft part of its Cuntz semigroup. Under a mild additional assumption, we show that every Cuntz class dominates a (unique) largest soft Cuntz class. This defines a retract from the Cuntz semigroup onto its soft part, and it follows that the covering dimensions of these semigroups differ by at most $1$.

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arXiv2023

Fraïssé Theory for Cuntz semigroups

We introduce a Fraïssé theory for abstract Cuntz semigroups akin to the theory of Fraïssé categories developed by Kubiś. In particular, we show that any (Cuntz) Fraïssé category has a unique Fraïssé limit which is both universal and homogeneous. We also give several examples of such categories and compute their Fraïssé limits. During our investigations, we develop a general theory of Cauchy sequences and intertwinings in the category Cu.

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arXiv2023

The Cuntz semigroup of a ring

For any ring $R$, we introduce an invariant in the form of a partially ordered abelian semigroup $\mathrm{S}(R)$ built from an equivalence relation on the class of countably generated projective modules. We call $\mathrm{S}(R)$ the Cuntz semigroup of the ring $R$. This construction is akin to the manufacture of the Cuntz semigroup of a C*-algebra using countably generated Hilbert modules. To circumvent the lack of a topology in a general ring $R$, we deepen our understanding of countably projective modules over $R$, thus uncovering new features in their direct limit decompositions, which in turn yields two equivalent descriptions of $\mathrm{S}(R)$. The Cuntz semigroup of $R$ is part of a new invariant $\mathrm{SCu}(R)$ which includes an ambient semigroup in the category of abstract Cuntz semigroups that provides additional information. We provide computations for both $\mathrm{S}(R)$ and $\mathrm{SCu}(R)$ in a number of interesting situations, such as unit-regular rings, semilocal rings, and in the context of nearly simple domains. We also relate our construcion to the Cuntz semigroup of a C*-algebra.

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arXiv2023

Nowhere scattered multiplier algebras

We study sufficient conditions under which a nowhere scattered C*-algebra $A$ has a nowhere scattered multiplier algebra $\mathcal{M}(A)$, that is, we study when $\mathcal{M}(A)$ has no nonzero, elementary ideal-quotients. In particular, we prove that a $σ$-unital C*-algebra $A$ of finite nuclear dimension, or real rank zero, or stable rank one with $k$-comparison, is nowhere scattered if and only if $\mathcal{M}(A)$ is.

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arXiv2023

Soft operators in C*-algebras

We say that a C*-algebra is soft if it has no nonzero unital quotients, and we connect this property to the Hjelmborg-Rørdam condition for stability and to property (S) of Ortega-Perera-Rørdam. We further say that an operator in a C*-algebra is soft if its associated hereditary subalgebra is, and we provide useful spectral characterizations of this concept. Of particular interest are C*-algebras that have an abundance of soft elements in the sense that every hereditary subalgebra contains an almost full soft element. We show that this property is implied by the Global Glimm Property, and that every C*-algebra with an abundance of soft elements is nowhere scattered. This sheds new light on the long-standing Global Glimm Problem of whether every nowhere scattered C*-algebra has the Global Glimm Property.

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arXiv2022

The Global Glimm Property

It is known that a C*-algebra with the Global Glimm Property is nowhere scattered (it has no elementary ideal-quotients), and the Global Glimm Problem asks if the converse holds. We provide a new approach to this long-standing problem by showing that a C*-algebra has the Global Glimm Property if and only if it is nowhere scattered and its Cuntz semigroup is ideal-filtered (the Cuntz classes generating a given ideal are downward directed) and has property (V) (a weak form of being sup-semilattice ordered). We show that ideal-filteredness and property (V) are automatic for C*-algebras that have stable rank one or real rank zero, thereby recovering the solutions to the Global Glimm Problem in these cases. We also use our approach to solve the Global Glimm Problem for new classes of C*-algebras.

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