arXiv subjects

Joan Bosa

Publications and source records attributed to Joan Bosa.

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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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

The dynamical Cuntz semigroup and ideal-free quotients of Cuntz semigroups

We develop a theory of general quotients for W- and Cu-semigroups beyond the case of quotients by ideals. To this end, we introduce the notion of a normal pair, which allows us to take quotients of W-semigroups in a similar way as normal subgroups arise as kernels of group homomorphisms. We use this to define the dynamical Cuntz semigroup as the universal object induced from an action of a group G on a W-semigroup. In the C*-algebraic framework, under mild assumptions, the universality of this dynamical invariant helps us tap into the structure of the Cuntz semigroup of crossed product C*-algebras.

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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

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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arXiv2022

The nuclear dimension of $\mathcal O_\infty$-stable $C^*$-algebras

We show that every nuclear $\mathcal O_\infty$-stable *-homomorphism with a separable exact domain has nuclear dimension at most 1. In particular separable, nuclear, $\mathcal O_\infty$-stable C*-algebras have nuclear dimension 1. We also characterise when $\mathcal O_\infty$-stable C*-algebras have finite decomposition rank in terms of quasidiagonality and primitive-ideal structure, and determine when full $\mathcal O_2$-stable *-homomorphisms have nuclear dimension 0.

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arXiv2021

The type semigroup, comparison and almost finiteness for ample groupoids

We prove that a minimal second countable ample groupoid has dynamical comparison if and only if its type semigroup is almost unperforated. Moreover, we investigate to what extent a not necessarily minimal almost finite groupoid has an almost unperforated type semigroup. Finally, we build a bridge between coarse geometry and topological dynamics by characterizing almost finiteness of the coarse groupoid in terms of a new coarsely invariant property for metric spaces, which might be of independent interest in coarse geometry. As a consequence, we are able to construct new examples of almost finite principal groupoids lacking other desirable properties, such as amenability or even a-T-menability. This behaviour is in stark contrast to the case of principal transformation groupoids associated to group actions.

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arXiv2021

Stable Elements and Property (S)

We study the relation (and differences) between stability and Property (S) in the simple and stably finite framework. This leads us to characterize stable elements in terms of its support, and study these concepts from different sides : hereditary subalgebras, projections in the multiplier algebra and order properties in the Cuntz semigroup. We use these approaches to show both that cancellation at infinity on the Cuntz semigroup just holds when its Cuntz equivalence is given by isomorphism at the level of Hilbert right-modules, and that different notions as Regularity, $ω$-comparison, Corona Factorization Property, property R, etc.. are equivalent under mild assumptions.

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arXiv2020

The Realization Problem for Finitely Generated Refinement Monoids

We show that every finitely generated conical refinement monoid can be represented as the monoid $\mathcal V(R)$ of isomorphism classes of finitely generated projective modules over a von Neumann regular ring $R$. To this end, we use the representation of these monoids provided by adaptable separated graphs. Given an adaptable separated graph $(E, C)$ and a field $K$, we build a von Neumann regular $K$-algebra $Q_K (E, C)$ and show that there is a natural isomorphism between the separated graph monoid $M(E, C)$ and the monoid $\mathcal V(Q_K (E, C))$.

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arXiv2020

Strict comparison for $C^*$-algebras arising from Almost finite groupoids

In this paper we show that for an almost finite minimal ample groupoid $G$, its reduced $\mathrm{C}^*$-algebra $C_r^*(G)$ has real rank zero and strict comparison even though $C_r^*(G)$ may not be nuclear in general. Moreover, if we further assume $G$ being also second countable and non-elementary, then its Cuntz semigroup ${\rm Cu}(C_r^*(G))$ is almost divisible and ${\rm Cu}(C_r^*(G))$ and ${\rm Cu}(C_r^*(G)\otimes \mathcal{Z})$ are canonically order-isomorphic, where $\mathcal{Z}$ denotes the Jiang-Su algebra.

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arXiv2020

The groupoids of adaptable separated graphs and their type semigroup

Given an adaptable separated graph, we construct an associated groupoid and explore its type semigroup. Specifically, we first attach to each adaptable separated graph a corresponding semigroup, which we prove is an $E^*$-unitary inverse semigroup. As a consequence, the tight groupoid of this semigroup is a Hausdorff étale groupoid. We show that this groupoid is always amenable, and that the type semigroups of groupoids obtained from adaptable separated graphs in this way include all finitely generated conical refinement monoids. The first three named authors will utilize this construction in forthcoming work to solve the Realization Problem for von Neumann regular rings, in the finitely generated case.

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arXiv2016

Reconstruction of graded groupoids from graded Steinberg algebras

We show how to reconstruct a graded ample Hausdorff groupoid with topologically principal neutrally graded component from the ring structure of its graded Steinberg algebra over any commutative integral domain with 1, together with the embedding of the canonical abelian subring of functions supported on the unit space. We deduce that diagonal-preserving ring isomorphism of Leavitt path algebras implies $C^*$-isomorphism of $C^*$-algebras for graphs $E$ and $F$ in which every cycle has an exit.

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arXiv2016

Covering dimension of C*-algebras and 2-coloured classification

We introduce the concept of finitely coloured equivalence for unital *-homomorphisms between C*-algebras, for which unitary equivalence is the 1-coloured case. We use this notion to classify *-homomorphisms from separable, unital, nuclear C*-algebras into ultrapowers of simple, unital, nuclear, Z-stable C*-algebras with compact extremal trace space up to 2-coloured equivalence by their behaviour on traces; this is based on a 1-coloured classification theorem for certain order zero maps, also in terms of tracial data. As an application we calculate the nuclear dimension of non-AF, simple, separable, unital, nuclear, Z-stable C*-algebras with compact extremal trace space: it is 1. In the case that the extremal trace space also has finite topological covering dimension, this confirms the remaining open implication of the Toms-Winter conjecture. Inspired by homotopy-rigidity theorems in geometry and topology, we derive a "homotopy equivalence implies isomorphism" result for large classes of C*-algebras with finite nuclear dimension.

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arXiv2016

Comparison Properties of the Cuntz semigroup and applications to C*-algebras

We study comparison properties in the category Cu aiming to lift results to the C*-algebraic setting. We introduce a new comparison property and relate it to both the CFP and $ω$-comparison. We show differences of all properties by providing examples, which suggest that the corona factorization property for C*-algebras might allow for both finite and infinite projections. In addition, we show that Rørdam's simple, nuclear C*-algebra with a finite and an infinite projection does not have the CFP.

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arXiv2016

A Bivariant Theory for the Cuntz Semigroup

We introduce a bivariant version of the Cuntz semigroup as equivalence classes of order zero maps generalizing the ordinary Cuntz semigroup. The theory has many properties formally analogous to KK-theory including a composition product. We establish basic properties, like additivity, stability and continuity, and study categorical aspects in the setting of local C*-algebras. We determine the bivariant Cuntz semigroup for numerous examples such as when the second algebra is a Kirchberg algebra, and Cuntz homology for compact Hausdorff spaces which provides a complete invariant. Moreover, we establish identities when tensoring with strongly self-absorbing C*-algebras. Finally, we show that the bivariant Cuntz semigroup of the present work can be used to classify all unital and stably finite C*-algebras.

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arXiv2015

Open Projections and Suprema in the Cuntz Semigroup

We provide a new and concise proof of the existence of suprema in the Cuntz semigroup using the open projection picture of the Cuntz semigroup initiated by Ortega, Rordam and Thiel. Our argument is based on the observation that the supremum of a countable set of open projections in the bidual of a C*-algebra A is again open and corresponds to the generated hereditary sub-C*-algebra of A.

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arXiv2013

Geometric Structure of Dimension Functions of Certain Continuous Fields

In this paper we study structural properties of the Cuntz semigroup and its functionals for continuous fields of C*-algebras over finite dimensional spaces. In a variety of cases, this leads to an answer to a conjecture posed by Blackadar and Handelman. Enroute to our results, we determine when the stable rank of continuous fields of C*-algebras over one dimensional spaces is one.

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arXiv2012

The Cuntz semigroup of continuous fields

In this paper we describe the Cuntz semigroup of continuous fields of C$^*$-algebras over one dimensional spaces whose fibers have stable rank one and trivial $K_1$ for each closed, two-sided ideal. This is done in terms of the semigroup of global sections on a certain topological space built out of the Cuntz semigroups of the fibers of the continuous field. When the fibers have furthermore real rank zero, and taking into account the action of the space, our description yields that the Cuntz semigroup is a classifying invariant if and only if so is the sheaf induced by the Murray-von Neumann semigroup.

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