Searcharxiv⌕ Search

arXiv subjects

Eduardo Scarparo

Publications and source records attributed to Eduardo Scarparo.

15 recordsLinked to original sources
arXiv2026

Invariant measures and traces on groupoid $\boldsymbol{\mathrm{C}^\ast}$-algebras

We provide sufficient conditions for the existence of a trace on the essential $\mathrm{C}^\ast$-algebra of a (not necessarily Hausdorff) étale groupoid $G$ which extends an invariant measure $μ$ on the unit space of $G$. In particular, it suffices for the isotropy groups of $G$ to be amenable, or for $G$ to be essentially free with respect to $μ$. We also show that $G$ is essentially free with respect to an invariant measure $μ$ if and only if $μ$ extends to a unique trace on the full $\mathrm{C}^\ast$-algebra of $G$. We work in the generality of possibly infinite measures and, accordingly, possibly unbounded traces. Moreover, whenever possible, we state our results for twisted groupoids. As an application, we show that gauge-invariant algebras of finite-state self-similar groups admit a unique tracial state.

math.OA↗
arXiv2025

Maximal ideals of reduced group C*-algebras and Thompson's groups

Given a conditional expectation $P$ from a C*-algebra $B$ onto a C*-subalgebra $A$, we observe that induction of ideals via $P$, together with a map which we call co-induction, forms a Galois connection between the lattices of ideals of $A$ and $B$. Using properties of this Galois connection, we show that, given a discrete group $G$ and a stabilizer subgroup $G_x$ for the action of $G$ on its Furstenberg boundary, induction gives a bijection between the set of maximal co-induced ideals of $C^*(G_x)$ and the set of maximal ideals of $C^*_r(G)$. As an application, we prove that the reduced C*-algebra of Thompson's group $T$ has a unique maximal ideal. Furthermore, we show that, if Thompson's group $F$ is amenable, then $C^*_r(T)$ has infinitely many ideals.

math.OA↗
arXiv2023

A tracial characterization of Furstenberg's $\times p,\times q$ conjecture

We investigate almost minimal actions of abelian groups and their crossed products. As an application, given multiplicatively independent integers $p$ and $q$, we show that Furstenberg's $\times p,\times q$ conjecture holds if and only if the canonical trace is the only faithful extreme tracial state on the $C^*$-algebra of the group $\mathbb{Z}[\frac{1}{pq}]\rtimes\mathbb{Z}^2$. We also compute the primitive ideal space and K-theory of $C^*(\mathbb{Z}[\frac{1}{pq}]\rtimes\mathbb{Z}^2)$.

math.OA↗
arXiv2023

The AH conjecture for Cantor minimal dihedral systems

The AH conjecture relates the low-dimensional homology groups of a groupoid with the abelianization of its topological full group. We show that transformation groupoids of minimal actions of the infinite dihedral group on the Cantor set satisfy this conjecture. The proof uses Kakutani-Rokhlin partitions adapted to such systems.

math.GR↗
arXiv2023

$C^*$-irreducibility of commensurated subgroups

Given a commensurated subgroup $Λ$ of a group $Γ$, we completely characterize when the inclusion $Λ\leq Γ$ is $C^*$-irreducible and provide new examples of such inclusions. In particular, we obtain that $\rm{PSL}(n,\mathbb{Z})\leq\rm{PGL}(n,\mathbb{Q})$ is $C^*$-irreducible for any $n\in \mathbb{N}$, and that the inclusion of a $C^*$-simple group into its abstract commensurator is $C^*$-irreducible. The main ingredient that we use is the fact that the action of a commensurated subgroup $Λ\leqΓ$ on its Furstenberg boundary $\partial_FΛ$ can be extended in a unique way to an action of $Γ$ on $\partial_FΛ$. Finally, we also investigate the counterpart of this extension result for the universal minimal proximal space of a group.

math.OA↗
arXiv2022

A dichotomy for topological full groups

Given a minimal action $α$ of a countable group on the Cantor set, we show that the alternating full group $\mathsf{A}(α)$ is non-amenable if and only if the topological full group $\mathsf{F}(α)$ is $C^*$-simple. This implies, for instance, that the Elek-Monod example of non-amenable topological full group coming from a Cantor minimal $\mathbb{Z}^2$-system is $C^*$-simple.

math.GR↗
arXiv2022

Boundary maps and covariant representations

We extend applications of Furstenberg boundary theory to the study of $C^*$-algebras associated to minimal actions $Γ\!\curvearrowright\! X$ of discrete groups $Γ$ on locally compact spaces $X$. We introduce boundary maps on $(Γ,X)$-$C^*$-algebras and investigate their applications in this context. Among other results, we completely determine when $C^*$-algebras generated by covariant representations arising from stabilizer subgroups are simple. We also characterize the intersection property of locally compact $Γ$-spaces and simplicity of their associated crossed products.

math.OA↗
arXiv2021

Boundary maps, germs and quasi-regular representations

We investigate the tracial and ideal structures of $C^*$-algebras of quasi-regular representations of stabilizers of boundary actions. Our main tool is the notion of boundary maps, namely $Γ$-equivariant unital completely positive maps from $Γ$-$C^*$-algebras to $C(\partial_FΓ)$, where $\partial_FΓ$ denotes the Furstenberg boundary of a group $Γ$. For a unitary representation $π$ coming from the groupoid of germs of a boundary action, we show that there is a unique boundary map on $C^*_π(Γ)$. Consequently, we completely describe the tracial structure of the $C^*$-algebras $C^*_π(Γ)$, and for any $Γ$-boundary $X$, we completely characterize the simplicity of the $C^*$-algebras generated by the quasi-regular representations $λ_{Γ/Γ_x}$ associated to stabilizer subgroups $Γ_x$ for any $x\in X$. As an application, we show that the $C^*$-algebra generated by the quasi-regular representation $λ_{T/F}$ associated to Thompson's groups $F\leq T$ does not admit traces and is simple.

math.OA↗
arXiv2021

Relatively amenable actions of Thompson's groups

We investigate the notion of relatively amenable topological action and show that the action of Thompson's group $T$ on $S^1$ is relatively amenable with respect to Thompson's group $F$. We use this to conclude that $F$ is exact if and only if $T$ is exact. Moreover, we prove that the groupoid of germs of the action of $T$ on $S^1$ is Borel amenable.

math.GR↗
arXiv2021

Almost finiteness and homology of certain non-free actions

We show that Cantor minimal $\mathbb{Z}\rtimes\mathbb{Z}_2$-systems and essentially free amenable odometers are almost finite. We also compute the homology groups of Cantor minimal $\mathbb{Z}\rtimes\mathbb{Z}_2$-systems and show that the associated transformation groupoids satisfy the HK conjecture if and only if the action is free.

math.OA↗
arXiv2020

A torsion-free algebraically C*-unique group

Let $p$ and $q$ be multiplicatively independent integers. We show that the complex group ring of $\mathbb{Z}[\frac{1}{pq}]\rtimes\mathbb{Z}^2$ admits a unique $\mathrm{C}^*$-norm. The proof uses a characterization, due to Furstenberg, of closed $\times p-$ and $\times q-$invariant subsets of $\mathbb{T}$.

math.OA↗
arXiv2019

$C^*$-simplicity and representations of topological full groups of groupoids

Given an ample groupoid $G$ with compact unit space, we study the canonical representation of the topological full group $[[G]]$ in the full groupoid $C^*$-algebra $C^*(G)$. In particular, we show that the image of this representation generates $C^*(G)$ if and only if $C^*(G)$ admits no tracial state. The techniques that we use include the notion of groups covering groupoids. As an application, we provide sufficient conditions for $C^*$-simplicity of certain topological full groups, including those associated with topologically free and minimal actions of non-amenable and countable groups on the Cantor set.

math.OA↗
arXiv2019

On the $C^*$-algebra generated by the Koopman representation of a topological full group

Let $(X,T,μ)$ be a Cantor minimal sytem and $[[T]]$ the associated topological full group. We analyze $C^*_π([[T]])$, where $π$ is the Koopman representation attached to the action of $[[T]]$ on $(X,μ)$. Specifically, we show that $C^*_π([[T]])=C^*_π([[T]]')$ and that the kernel of the character $τ$ on $C^*_π([[T]])$ coming from weak containment of the trivial representation is a hereditary $C^*$-subalgebra of $C(X)\rtimes\mathbb{Z}$. Consequently, $\kerτ$ is stably isomorphic to $C(X)\rtimes\mathbb{Z}$, and $C^*_π([[T]]')$ is not AF. We also prove that if $G$ is a finitely generated, elementary amenable group and $C^ *(G)$ has real rank zero, then $G$ is finite.

math.OA↗
arXiv2019

Homology of odometers

We compute the homology groups of transformation groupoids associated with odometers and show that certain infinite dihedral group odometers give rise to counterexamples to the HK conjecture, which relates the homology of an essentially principal, minimal, ample groupoid $G$ with the K-theory of $C^*_r(G)$. We also show that transformation grupoids of odometers satisfy the AH conjecture.

math.OA↗