arXiv subjects

Alcides Buss

Publications and source records attributed to Alcides Buss.

At least 19 recordsLinked to original sources

arXiv2026

Von Neumann algebras as reduced twisted groupoid $C^*$-algebras

We characterize the von Neumann algebras that are isomorphic, as $C^*$-algebras, to reduced twisted $C^*$-algebras of locally compact Hausdorff groupoids equipped with continuous Haar systems of full support. They are exactly the subhomogeneous von Neumann algebras, equivalently finite products of matrix algebras over abelian von Neumann algebras. The groupoid can always be chosen compact, principal, and \'etale, with trivial twist and counting Haar system. We also prove that, for a groupoid in this class, the full or reduced twisted algebra is unital if and only if the groupoid is \'etale with compact unit space. This criterion reduces the classification for general Haar systems to the \'etale case. A further obstruction comes from controlled propagation, defined through faithful representations into the norm closure of uniformly sparse matrices. Every reduced twisted \'etale groupoid algebra and its Borel completion have controlled propagation, including for non-Hausdorff groupoids with locally compact Hausdorff unit space. For every infinite-dimensional Hilbert space $H$, every $*$-homomorphism from a nonzero quotient of $B(H)$ into an algebra with controlled propagation is zero. In particular, neither $B(H)$ nor the Calkin algebra embeds into any of these reduced or Borel groupoid algebras. We also prove that every von Neumann algebra with controlled propagation is finite. No separability or countability assumptions are required.

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arXiv2026

Dynamics in large scale geometry

We investigate the large scale geometry of certain metric spaces through the lens of dynamics. Our approach establishes a close connection between large scale dynamical phenomena and operator algebras by characterizing various large scale dynamic behaviors in terms of GNS representations of the uniform Roe algebras arising from natural canonical states. Our dynamical systems are given by the Stone-\v{C}ech boundary of metric spaces together with their inverse semigroup of partial translations. This defines a space of orbits and we characterize Hausdorffness and $T_1$-ness of this space by the failure of coarse embeddability of certain metric spaces. Surprisingly, while the orbit space has very weak separation properties, we show that it satisfies a certain ''localized version'' of Urysohn's lemma. We show that the topology of the space of orbits and quasi-orbits are given by the space of irreducible representations of uniform Roe algebras and by the space of their primitive ideals, respectively. As a highlight of the theory developed herein, we provide classes of spaces such that the prime ideals of their uniform Roe algebras are primitive. This is the case for instance of spaces whose orbit space is $T_1$.

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arXiv2026

Rapid decay and localizability for Fell bundles over etale Groupoids

We introduce a notion of the Rapid Decay Property (RDP) for Fell bundles over locally compact Hausdorff \'etale groupoids, extending earlier rapid decay theories for \'etale groupoids and twists. Our approach yields analytic control on convolution norms and leads to the existence of dense Schwartz-type $*$-subalgebras of the reduced cross-sectional $C^*$-algebra $C_r^*(E)$. As an application, we obtain approximation results showing that, under suitable hypotheses, sections of $C_r^*(E)$ with support contained in an open subset $U\subseteq G$ can be approximated in the reduced norm by compactly supported sections supported inside $U$. In this sense, the Rapid Decay Property provides an analytic mechanism leading to a form of localizability for Fell bundles. We also investigate the relationship between RDP, polynomial growth, and dynamical systems. We show that Fell bundles over groupoids with polynomial growth naturally satisfy the RDP. Furthermore, for a transformation groupoid $G=\Gamma\ltimes_\theta X$ associated with a partial action, we prove that RDP for a Fell bundle over $G$ is equivalent to RDP for a naturally associated Fell bundle over the discrete group $\Gamma$. Finally, we apply these tools to Deaconu-Renault groupoids. By realizing them as partial crossed products of free groups, we show that the presence of persistent branching forces exponential growth, completely obstructing the RDP. This provides a striking illustration of a system where the acting group has RDP, but the associated groupoid fails to inherit it, fully clarifying the boundary between the group and groupoid theories.

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arXiv2026

On groupoids beyond partial actions, inner amenability, and models for Kirchberg algebras

We construct the first explicit examples of locally compact Hausdorff \'etale groupoids that are not inner amenable and that do not arise as transformation groupoids associated to partial actions of discrete groups. This answers questions of Anantharaman--Delaroche and Exel. Our examples include all Higson--Lafforgue--Skandalis groupoids associated to non-amenable residually finite groups, as well as their principal variants constructed by Alekseev--Finn--Sell. These can be chosen to be second countable, ample, and in the latter case even principal. We also show that large classes of Deaconu--Renault groupoids with connected unit space do not arise from partial actions of discrete groups, including cases whose $C^*$-algebras are Kirchberg algebras in the UCT class. We contrast this with the totally disconnected case by giving ample transformation groupoid models for all unital Kirchberg algebras in the UCT class as well as many higher rank graph algebras. Finally, we characterize precisely when coarse groupoids arise from partial actions of discrete groups in terms of coarse embeddings into groups.

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arXiv2026

A universal property for groupoid C*-algebras. II. Fell bundles

We define possibly unsaturated, upper semicontinuous Fell bundles over Hausdorff, locally compact groupoids and establish a universal property for representations of their full section C*-algebras on Hilbert modules over arbitrary C*-algebras. Based on this, we prove that the full section C*-algebra is functorial and exact, and we define a quasi-orbit space and a quasi-orbit map. We deduce and extend Renault's Integration and Disintegration Theorems to general Fell bundles using our universal property.

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arXiv2026

B(H) is not a twisted groupoid C*-algebra

We show that $B(H)$ for an infinite dimensional Hilbert space $H$ cannot be realized as the reduced twisted $C^*$-algebra of any locally compact Hausdorff \'etale groupoid. The proof is based on the canonical conditional expectation $$C_r^*(G,\Sigma)\to C_0(G^{(0)})$$ and a structural analysis of the resulting diagonal subalgebra inside $B(H)$. We show that this diagonal must be an atomic abelian von Neumann algebra, and then exclude both possibilities for its spectrum. If the unit space is finite, one obtains a tracial state on $C_r^*(G,\Sigma)$, which is impossible for $B(H)$. If it is infinite, the groupoid structure forces a block-sparsity phenomenon for compactly supported sections, which is incompatible with $B(H)$. This provides the first examples of $C^*$-algebras that cannot be realized as reduced twisted \'etale groupoid $C^*$-algebras.

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arXiv2025

W*-Amenability for Fell bundles over discrete groups

We investigate amenability for $W^*$-Fell bundles over a discrete group $G$, with a focus on its characterization via approximation properties and conditional expectations. Building on the notion of $W^*$-amenability, we construct an enlarged $W^*$-Fell bundle analogous to $\ell^\infty(G, M)$ for a group action $G$ on a von Neumann algebra $M$, and relate amenability to the existence of suitable conditional expectations at both the bundle and crossed-product levels. Our results unify and extend several approaches to amenability for noncommutative dynamical systems. As applications of our methods, we prove that amenability of Fell bundles passes to restrictions to subgroups and that a Fell bundle over a group $G$ is amenable if and only if both its restriction to a normal subgroup $H \trianglelefteq G$ and the associated quotient Fell bundle over $G/H$ are amenable. This provides a powerful structural tool that extends classical permanence results for group amenability to the setting of \Wstar Fell bundles and also \cstar algebraic Fell bundles. We also discuss how Fell bundles and their amenability interact with group coactions on $C^*$-algebras and von Neumann algebras.

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arXiv2025

The Leavitt inverse semigroup of a separated graph

We introduce and study a new inverse semigroup associated to a separated graph $(E,C)$, which we call the \emph{Leavitt inverse semigroup}. This semigroup is obtained as a quotient of the separated graph inverse semigroup $\mathcal{S}(E,C)$, introduced in our previous paper [9], and it provides a canonical inverse semigroup model for the tame Leavitt path algebra $\mathcal{L}_K^\mathrm{ab}(E,C)$ over a commutative unital ring $K$. Our first main result describes the Leavitt inverse semigroup $\mathcal{LI}(E,C)$ as a restricted semidirect product of the free group on the edges of $E$ acting partially on a certain semilattice, which is isomorphic to the semilattice of idempotents of $\mathcal{LI}(E,C)$. This description, given in terms of Leavitt--Munn trees, yields a normal form for the elements of $\mathcal{LI}(E,C)$. We obtain a normal form for elements of $\mathcal{L}_K^\mathrm{ab}(E,C)$, leading to explicit linear bases for $\mathcal{L}_K^\mathrm{ab}(E,C)$. Building on this and on the structural properties of $\mathcal{LI} (E,C)$, we prove that the natural homomorphism from $\mathcal{LI}(E,C)$ to $\mathcal{L}_K^\mathrm{ab}(E,C)$ is injective, so that $\mathcal{LI}(E,C)$ embeds as the inverse semigroup generated by the canonical partial isometries in $\mathcal{L}_K^\mathrm{ab}(E,C)$. Further applications include the determination of natural bases of the kernel $\mathcal Q$ of the natural map from the tame Cohn algebra $\mathcal{C}_K^\mathrm{ab} (E,C)$ to the tame Leavitt path algebtra $\mathcal{L}_K^\mathrm{ab} (E,C)$, the computation of the socle, and a characterization of the isolated points of the spectrum. Several examples, such as the Cuntz separated graph and free separations, are discussed to illustrate the theory.

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arXiv2025

Characterizations of amenability for noncommutative dynamical systems and Fell bundles

We resolve key open questions regarding approximation properties and their permanence for Fell bundles over locally compact groups. Specifically, we establish the equivalence between the B\'edos--Conti approximation property (BCAP) and the Exel--Ng positive approximation property (AP), completely removing the necessity of assuming nuclearity on the unit fiber. To overcome the obstructions present in general Fell bundles (such as the lack of spatial arguments and exactness), we introduce a diagonal maximal tensor product $\otimes^d_{\max}$. We prove that a Fell bundle $\mathcal{A}$ has the AP if and only if $\mathcal{A} \otimes^d_{\max} \mathcal{B}$ has the weak containment property (wcp) for every Fell bundle $\mathcal{B}$. For $C^*$-dynamical systems, this yields a characterization of amenability that was known to hold under exactness assumptions. Furthermore, this tensorial machinery allows us to establish highly non-trivial permanence properties for the AP, including passage to restrictions over closed subgroups and partial quotients by normal subgroups. We also provide applications concerning the nuclearity of full and reduced cross-sectional $C^*$-algebras.

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arXiv2025

Operator algebras over the p-adic integers -- II

We continue the study of operator algebras over the $p$-adic integers, initiated in our previous work [1]. In this sequel, we develop further structural results and provide new families of examples. We introduce the notion of $p$-adic von Neumann algebras, and analyze those with trivial center, that we call ''factors''. In particular we show that ICC groups provide examples of factors. We then establish a characterization of $p$-simplicity for groupoid operator algebras, showing its relation to effectiveness and minimality. A central part of the paper is devoted to a $p$-adic analogue of the GNS construction, leading to a representation theorem for Banach $^*$-algebras over $\mathbb{Z}_p$. As applications, we exhibit large classes of $p$-adic operator algebras, including residually finite-rank algebras and affinoid algebras with the spectral norm. Finally, we investigate the $K$-theory of $p$-adic operator algebras, including the computation of homotopy analytic $K$-theory of continuous $\mathbb{Z}_p$-valued functions on a compact Hausdorff space and the analytic (non-homotopy invariant) $K$-theory of certain $p$-adically complete Banach algebras in terms of continuous $K$-theory. Together, these results extend the foundations of the emerging theory of $p$-adic operator algebras.

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arXiv2025

Fischer's approach to deformation of coactions

This paper explores a novel approach to the deformation of $C^*$-algebras via coactions of locally compact groups, emphasizing Fischer's construction in the context of maximal coactions. We establish a rigorous framework for understanding how deformations arise from group coactions, extending previous work by Bhowmick, Neshveyev, and Sangha. Using Landstad duality, we compare different deformation procedures, demonstrating their equivalence and efficiency in constructing twisted versions of given $C^*$-algebras. Our results provide deeper insights into the interplay between exotic crossed products, coaction duality, and operator algebra deformations, offering a unified perspective for further generalizations.

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arXiv2025

Essential groupoid amenability and nuclearity of groupoid C*-algebras

We give an alternative construction of the essential $C^*$-algebra of an \'etale groupoid, along with an ``amenability'' notion for such groupoids that is implied by the nuclearity of this essential $C^*$-algebra. In order to do this we first introduce a maximal version of the essential $C^*$-algebra, and prove that every function with dense co-support can only be supported on the set of ``dangerous'' arrows. We then introduce an essential amenability condition for a groupoid, which is (strictly) weaker than its (topological) amenability. As an application, we describe the Bruce-Li algebras arising from algebraic actions of cancellative semigroups as exotic essential $C^*$-algebras.

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arXiv2024

Flows on uniform Roe algebras

For a uniformly locally finite metric space $(X, d)$, we investigate \emph{coarse} flows on its uniform Roe algebra $\mathrm{C}^*_u(X)$, defined as one-parameter groups of automorphisms whose differentiable elements include all partial isometries arising from partial translations on $X$. We first show that any flow $\sigma$ on $\mathrm{C}^*_u(X)$ corresponds to a (possibly unbounded) self-adjoint operator $h$ on $\ell_2(X)$ such that $\sigma_t(a) = e^{ith} a e^{-ith}$ for all $t \in \mathbb{R}$, allowing us to focus on operators $h$ that generate flows on $ \mathrm{C}^*_u (X)$. Assuming Yu's property A, we prove that a self-adjoint operator $h$ on $\ell_2(X)$ induces a coarse flow on $\mathrm{C}^*_u(X)$ if and only if $h$ can be expressed as $h = a + d$, where $a \in \mathrm{C}^*_u(X)$ and $d$ is a diagonal operator with entries forming a coarse function on $X$. We further study cocycle equivalence and cocycle perturbations of coarse flows, showing that, under property A, any coarse flow is a cocycle perturbation of a diagonal flow. Finally, for self-adjoint operators $h$ and $k$ that induce coarse flows on $\mathrm{C}^*_u(X)$, we characterize conditions under which the associated flows are either cocycle perturbations of each other or cocycle conjugate. In particular, if $h - k$ is bounded, then the flow induced by $h$ is a cocycle perturbation of the flow induced by $k$.

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arXiv2024

Fourier--Stieltjes category for twisted groupoid actions

We extend the theory of Fourier--Stieltjes algebras to the category of twisted actions by \'etale groupoids on arbitrary C*-bundles, generalizing theories constructed previously by B\'{e}dos and Conti for twisted group actions on unital C*-algebras, and by Renault and others for groupoid C*-algebras, in each case motivated by the classical theory of Fourier--Stieltjes algebras of discrete groups. To this end we develop a toolbox including, among other things, a theory of multiplier C*-correspondences, multiplier C*-correspondence bundles, Busby--Smith twisted groupoid actions, and the associated crossed products, equivariant representations and Fell's absorption theorems. For a fixed \'etale groupoid $G$ a Fourier--Stieltjes multiplier is a family of maps acting on fibers, arising from an equivariant representation. It corresponds to a certain fiber-preserving strict completely bounded map between twisted full (or reduced) crossed products. We establish a KSGNS-type dilation result which shows that the correspondence above restricts to a bijection between positive-definite multipliers and a particular class of completely positive maps. Further we introduce a subclass of Fourier multipliers, that enjoys a natural absorption property with respect to Fourier--Stieltjes multipliers and gives rise to `reduced to full' multiplier maps on crossed products. Finally we provide several applications of the theory developed, for example to the approximation properties, such as weak containment or nuclearity, of the crossed products and actions in question, and discuss outstanding open problems.

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arXiv2024

Inverse semigroups of separated graphs and associated algebras

In this paper we introduce an inverse semigroup $\mathcal{S}(E,C)$ associated to a separated graph $(E,C)$ and describe its internal structure. In particular we show that it is strongly $E^*$-unitary and can be realized as a partial semidirect product of the form $\mathcal{Y}\rtimes\mathbb{F}$ for a certain partial action of the free group $\mathbb{F}=\mathbb{F}(E^1)$ on the edges of $E$ on a semilattice $\mathcal{Y}$ realizing the idempotents of $\mathcal{S}(E,C)$. In addition we also describe the spectrum as well as the tight spectrum of $\mathcal{Y}$. We then use the inverse semigroup $\mathcal{S}(E,C)$ to describe several "tame" algebras associated to $(E,C)$, including its Cohn algebra, its Leavitt-path algebra, and analogues in the realm of $C^*$-algebras, like the tame $C^*$-algebra $\mathcal{O}(E,C)$ and its Toeplitz extension $\mathcal{T}(E,C)$, proving that these algebras are canonically isomorphic to certain algebras attached to $\mathcal{S}(E,C)$. Our structural results on $\mathcal{S}(E,C)$ imply that these algebras can be realized as partial crossed products, revealing a great portion of their structure.

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arXiv2024

Operator algebras over the p-adic integers

We introduce $p$-adic operator algebras, which are nonarchimedean analogues of $C^*$-algebras. We demonstrate that various classical examples of operator algebras - such as group(oid) $C^*$-algebras - have nonarchimedean counterparts. The category of $p$-adic operator algebras exhibits similar properties to those of the category of real and complex $C^*$-algebras, featuring limits, colimits, tensor products, crossed products and an enveloping construction permitting us to construct $p$-adic operator algebras from involutive algebras over $\mathbb{Z}_p$. In several cases of interest, the enveloping algebra construction recovers the $p$-adic completion of the underlying $\mathbb{Z}_p$-algebra. We then discuss an analogue of topological $K$-theory for Banach $\mathbb{Z}_p$-algebras, and compute it in basic examples such as the $p$-adic Cuntz algebra and rotation algebras. Finally, for a large class of $p$-adic operator algebras, we show that our $K$-theory coincides with the reduction mod $p$ of Quillen's algebraic $K$-theory.

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arXiv2024

Deformation of Fell bundles

In this paper we study deformations of $C^*$-algebras that are given as cross-sectional $C^*$-algebras of Fell bundles over locally compact groups $G$. Our deformation comes from a direct deformation of the Fell bundles via certain parameters, like automorphisms of the Fell bundle, group cocycles, or central group extensions of $G$ by the circle group $\mathbb{T}$, and then taking cross-sectional algebras of the deformed Fell bundles. We then show that this direct deformation method is equivalent to the deformation via the dual coactions by similar parameters as studied previously in [4,7].

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arXiv2023

A new approach to deformation of C*-algebras via coactions

We revisit the procedure of deformation of $C^*$-algebras via coactions of locally compact groups and extend the methods to cover deformations for maximal, reduced, and exotic coactions for a given group $G$ and circle-valued Borel $2$-cocycles on $G$. In the special case of reduced (or normal) coactions our deformation method substantially differs from -- but turns out to be equivalent to -- the ones used by previous authors, specially those given by Bhowmick, Neshveyev, and Sangha in [7]. Our approach yields all expected results, like a good behaviour of deformations under nuclearity, continuity of fields of $C^*$-algebras and $K$-theory invariance under mild conditions.

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