arXiv subjects

Damián Ferraro

Publications and source records attributed to Damián Ferraro.

12 recordsLinked to original sources

arXiv2026

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 Bédos--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

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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arXiv2023

Cross-sectional C*-algebras associated to subgroups

Given a Fell bundle $\mathcal{B}=\{B_t\}_{t\in G}$ over a locally compact and Hausdorff group $G$ and a closed subgroup $H\subset G,$ we construct quotients $C^*_{H\uparrow \mathcal{B}}(\mathcal{B})$ and $C^*_{H\uparrow G}(\mathcal{B})$ of the full cross-sectional C*-algebra $C^*(\mathcal{B})$ analogous to Exel-Ng's reduced algebras $C^*_{\mathop{\rm r}}(\mathcal{B})\equiv C^*_{\{e\}\uparrow \mathcal{B}}(\mathcal{B})$ and $C^*_R(\mathcal{B})\equiv C^*_{\{e\}\uparrow G}(\mathcal{B}).$ An absorption principle, similar to Fell's one, is used to give conditions on $\mathcal{B}$ and $H$ (e.g. $G$ discrete and $\mathcal{B}$ saturated, or $H$ normal) ensuring $C^*_{H\uparrow \mathcal{B}}(\mathcal{B})=C^*_{H\uparrow G}(\mathcal{B}).$ The tools developed here enable us to show that if the normalizer of $H$ is open in $G$ and $\mathcal{B}_H:=\{B_t\}_{t\in H}$ is the reduction of $\mathcal{B}$ to $H,$ then $C^*(\mathcal{B}_H)=C^*_{\mathop{\rm r}}(\mathcal{B}_H)$ if and only if $C^*_{H\uparrow \mathcal{B}}(\mathcal{B})=C^*_{\mathop{\rm r}}(\mathcal{B});$ the last identification being implied by $C^*(\mathcal{B})=C^*_{\mathop{\rm r}}(\mathcal{B}).$ We also prove that if $G$ is inner amenable and $C^*_{\mathop{\rm r}}(\mathcal{B})\otimes_{\max} C^*_{\mathop{\rm r}}(G)=C^*_{\mathop{\rm r}}(\mathcal{B})\otimes C^*_{\mathop{\rm r}}(G),$ then $C^*(\mathcal{B})=C^*_{\mathop{\rm r}}(\mathcal{B}).$

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arXiv2022

Induction, absorption and weak containment of -representations of Banach -algebraic bundles

Given a Fell bundle $\mathcal{B}=\{B_t\}_{t\in G}$ over a LCH group and a closed subgroup $H\subset G,$ we show that all the *-representations of $\mathcal{B}_H:=\{B_t\}_{t\in H}$ can be induced to *-representations of $\mathcal{B}$ by means of Fell's induction process; which we describe as induction via a *-homomorphism $q^{\mathcal{B}}_H\colon C^*(\mathcal{B})\to \mathbb{B}(X_{C^*(\mathcal{B}_H)}).$ The quotients $C^*_H(\mathcal{B}):=q^{\mathcal{B}}_H(C^*(\mathcal{B}))$ are intermediate to $C^*(\mathcal{B})= C^*_G(\mathcal{B})$ and $C^*_{r}(\mathcal{B})=C^*_{\{e\}}(\mathcal{B})$ because every inclusion of subgroups $H\subset K\subset G$ gives a unique quotient map $q^{\mathcal{B}}_{HK}\colon C^*_K(\mathcal{B})\to C^*_H(\mathcal{B})$ such that $q^{\mathcal{B}}_{HK}\circ q^{\mathcal{B}}_K=q^{\mathcal{B}}_H.$ All along the article we try to find conditions on $\mathcal{B},$ $G,\ H$ and $K$ (e.g. saturation, nuclearity or weak containment) that imply $q^{\mathcal{B}}_{HK}$ is faithful. One of our main tools is a blend of Fell's absorption principle (for saturated bundles) and a result of Exel and Ng for reduced cross sectional C*-algebras. We also show that given an imprimitivity system $\langle T,P\rangle$ for $\mathcal{B}$ over $G/H,$ if $H$ is open or has open normalizer in $G,$ then $T$ is weakly contained in a *-representation induced from $\mathcal{B}_H$ (even if $\mathcal{B}$ is not saturated). Given normal and closed subgroups of $G,$ $H\subset K,$ we construct a Fell bundle $\mathcal{C}$ over $G/K$ such that $C^*_r(\mathcal{C})=C^*_H(\mathcal{B}).$ We show that $q^{\mathcal{B}}_H$ is faithful if and only if both $q^{\mathcal{C}}_{\{e\}}$ and $q^{\mathcal{B}_K}_H$ are.

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arXiv2022

Nuclearity for partial crossed products by exact discrete groups

We study partial actions of exact discrete groups on C*-algebras. We show that the partial crossed product of a commutative C*-algebra by an exact discrete group is nuclear whenever the full and reduced partial crossed products coincide. This generalises a result by Matsumura in the context of global actions. In general, we prove that a partial action of an exact discrete group on a C*-algebra $A$ has Exel's approximation property if and only if the full and reduced partial crossed products associated to the diagonal partial action on $A\otimes_{\max} A^\mathrm{op}$ coincide. We apply our results to show that the reduced semigroup C*-algebra $\mathrm{C}^*_λ(P)$ of a submonoid of an exact discrete group is nuclear if the left regular representation on $\ell^2(P)$ is an isomorphism between the full and reduced C*-algebras. We also show that nuclearity is equivalent to the weak containment property in the case of C*-algebras associated to separated graphs.

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arXiv2021

Fixed point algebras for weakly proper Fell bundles

We define weakly proper Fell bundles and construct exotic fixed point algebras for such bundles. Three alternative constructions of such algebras are given. Under a kind of freeness condition, one of our constructions implies that every exotic cross sectional C*-algebra of a weakly proper Fell bundle is Morita equivalent to an exotic fixed point algebras. The other constructions are used to show that ours generalizes that of Buss and Echterhoff on weakly proper actions on C*-algebras. We also generalize to Fell bundles the fact that every C*-action which is proper in Kasparov's sense is amenable.

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arXiv2021

Amenability and approximation properties for partial actions and Fell bundles

Building on previous papers by Anantharaman-Delaroche (AD) we introduce and study the notion of AD-amenability for partial actions and Fell bundles over discrete groups. We prove that the cross-sectional C*-algebra of a Fell bundle is nuclear if and only if the underlying unit fibre is nuclear and the Fell bundle is AD-amenable. If a partial action is globalisable, then it is AD-amenable if and only if its globalisation is AD-amenable. Moreover, we prove that AD-amenability is preserved by (weak) equivalence of Fell bundles and, using a very recent idea of Ozawa and Suzuki, we show that AD-amenabity is equivalent to an approximation property introduced by Exel.

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arXiv2017

Morita enveloping Fell bundles

We introduce notions of weak and strong equivalence for non-saturated Fell bundles over locally compact groups and show that every Fell bundle is strongly (resp. weakly) equivalent to a semidirect product Fell bundle for a partial (resp. global) action. Equivalences preserve cross-sectional $C^*-$algebras and amenability. We use this to show that previous results on crossed products and amenability of group actions carry over to Fell bundles.

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arXiv2017

Equivalence of Fell bundles over groups

We give a notion of equivalence for Fell bundles over groups, not necessarily saturated nor separable, and show that equivalent Fell bundles have Morita-Rieffel equivalent cross-sectional $C^*$-algebras. Our notion is originated in the context of partial actions and their enveloping actions. The equivalence between two Fell bundles is implemented by a bundle of Hilbert bimodules with some extra structure. Suitable cross-sectional spaces of such a bundle turn out to be imprimitivity bimodules for the cross-sectional $C^*$-algebras of the involved Fell bundles. We show that amenability is preserved under this equivalence and, by means of a convenient notion of internal tensor product between Fell bundles, we show that equivalence of Fell bundles is an equivalence relation.

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arXiv2016

Applications of ternary rings to $C^*$-algebras

We show that there is a functor from the category of positive admissible ternary rings to the category of $*$-algebras, which induces an isomorphism of partially ordered sets between the families of $C^*$-norms on the ternary ring and its corresponding $*$-algebra. We apply this functor to obtain Morita-Rieffel equivalence results between cross sectional $C^*$-algebras of Fell bundles, and to extend the theory of tensor products of $C^*$-algebras to the larger category of full Hilbert $C^*$-modules. We prove that, like in the case of $C^*$-algebras, there exist maximal and minimal tensor products. As applications we give simple proofs of the invariance of nuclearity and exactness under Morita-Rieffel equivalence of $C^*$-algebras.

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arXiv2016

Construction of enveloping actions

We study the problem of constructing a globalization for partial actions on *-algebras, C*-algebras and Hilbert modules. For the first ones we give a necessary condition for the existence of a globalization and we prove this conditions is necessary and sufficient for C*-algebras. Using the linking algebra of a Hilbert module we translate this condition to the realm of partial action on Hilbert modules.

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arXiv2012

An Imprimitivity Theorem for Partial Actions

We define proper, free and commuting partial actions on upper semicontinuous bundles of $C^*-$algebras. With such, we construct the $C^*-$algebra induced by a partial action and a partial actions on that algebra. Using those action we give a generalization, to partial actions, of Raeburn's Symmetric Imprimitivity Theorem.

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