arXiv subjects

Eusebio Gardella

Publications and source records attributed to Eusebio Gardella.

At least 19 recordsLinked to original sources

arXiv2026

Equivariant $\mathcal{Z}$-stability for amenable actions

We prove that every amenable action of a countable discrete group on a separable, simple, unital, nuclear, $\mathcal{Z}$-stable C*-algebra absorbs the trivial action on $\mathcal{Z}$ up to cocycle conjugacy. The group may be nonamenable, as in Suzuki's recent examples, and no assumptions on outerness or on the tracial state space are needed. This extends OpenAI's result for amenable groups to amenable actions. Amenability of the action cannot be omitted in general, as examples of the author and Lupini show. More generally, we show that every tracially amenable action on a separable, unital, $\mathcal{Z}$-stable C*-algebra with traces admits invariant central copies of all matrix algebras, without simplicity or nuclearity. For amenable actions on simple nuclear algebras, equivariant property (SI) allows one to deduce $\mathcal{Z}$-stability from this. As applications, we obtain absorption of $\mathcal{Z}$ for intermediate subalgebras in the crossed product, and also pure infiniteness for outer amenable actions of nonamenable groups. The basic techniques involving Gaussian variables originate in OpenAI's preprint, and are also the main new tool in its proposed solution to the Toms--Winter conjecture. We extend these techniques in order to deal with the weight functions coming from amenability, and one of the main new ingredients is the fact that one can replace an amenable action by an auxiliary one for which the amenability weights are central. In order to make the probabilistic tools accessible to the community beyond what is used here, we include an appendix where the theory is developed assuming only an elementary background, and where the relevant estimates are proved in detail.

math.OA↗

arXiv2026

Isomorphisms of $\ell^1$-crossed products of topological systems

We give a complete description of the isometric isomorphisms between the $\ell^1$-crossed products of actions of discrete groups on nonempty compact Hausdorff spaces. In particular, two such crossed products are isometrically isomorphic as Banach algebras if and only if the actions are topologically isomorphic. It follows that $\ell^1(G\curvearrowright X)$ remembers not just $X$ but even the acting group and the action, and is not determined by the transformation groupoid $G\ltimes X$ alone. The proof recovers $C(X)$ from the hermitian elements and uses the geometry of $L^p$-spaces to identify the invertible isometries. We also obtain analogous results for measurable actions.

math.FA↗

arXiv2026

Classifiability of crossed products

This survey studies C*-algebraic crossed products arising from topological dynamical systems with an eye toward their classifiability in the sense of the Elliott program. We introduce the crossed product construction for actions by discrete groups in full generality, and then focus on commutative systems to establish some of the fundamental structural results: nuclearity of the crossed product is equivalent to amenability of the action, and in this setting simplicity is equivalent to the combination of minimality and topological freeness. With these properties in place, the only condition left to translate into dynamical terms is tensorial absorption of the Jiang-Su algebra $\mathcal{Z}$. We present the main dynamical tools known to obtain $\mathcal{Z}$-stability, both in the settings of amenable and nonamenable groups, and highlight the two conjectures that are believed to capture the full picture. The aim of the survey is to provide both an accessible entry point and a comprehensive reference on the classification of crossed products, describing the state of the art and a number of central open problems in the field.

math.OA↗

arXiv2026

Prime and semiprime Lie ideals in C*-algebras

Using the theory of Dixmier ideals developed in previous work, we show that every semiprime Lie ideal in a C*-algebra arises as the full normalizer subspace of a semiprime two-sided ideal. This leads to a concise description of all semiprime Lie ideals in terms of semiprime two-sided ideals, and an analogous description of prime Lie ideals in terms of prime two-sided ideals. For unital C*-algebras without characters, we obtain a natural bijection between (semi)prime two-sided ideals and (semi)prime Lie ideals, and it follows that a Lie ideal is fully noncentral if and only if it is not contained in any prime Lie ideal.

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arXiv2026

Embeddings of $L^p$-operator algebras

We study embeddings of $L^p$-operator algebras arising from (twis\-ted) étale groupoids, with particular emphasis on rigidity phenomena for $p\neq 2$. Our methods rely on a detailed analysis of core normalizers and their functorial behavior under algebra homomorphisms. Using the notion of actors between groupoids, we show that under natural hypotheses, embeddings between reduced $L^p$-groupoid algebras can be described entirely in terms of morphisms of the underlying groupoids. We further show that embeddings of $L^p$-groupoid algebras induce embeddings of the associated topological full groups. Our results provide new tools for studying embeddability questions in the $L^p$-setting, and are particularly helpful when ruling out the existence of embeddings. As applications, we obtain strong embeddability results both for spatial AF $L^p$-operator algebras and for tensor products of $L^p$-Cuntz algebras. For $p\not \in \{1,2\}$, a reduced $L^p$-groupoid algebra associated with a principal étale groupoid embeds into a spatial AF $L^p$-operator algebra if and only if the underlying groupoid is AF. In particular, and in contrast with classical results of Pimsner-Voiculescu, irrational $L^p$-noncommutative tori do not embed into spatial AF $L^p$-operator algebras for $p\neq 2$. Furthermore, if $p\neq 2$, there is no unital contractive homomorphism from $\mathcal{O}_2^p \otimes_p \mathcal{O}_2^p$ into $\mathcal{O}_2^p$, showing that there is no $L^p$-analog of Kirchberg's $\mathcal{O}_2$-embedding theorem.

math.FA↗

arXiv2025

Topological full groups, invertible isometries, and automorphisms of groupoid algebras

We show that the topological full group of a Hausdorff ample groupoid with compact unit space coincides with the group of homotopy classes of invertible isometries in pseudofunction algebras associated with the groupoid. Moreover, if the groupoid $\mathcal{G}$ is also effective, then we show that the group of (inner) automorphisms in pseudofunction algebras is a split extension of the automorphisms (respectively, the topological full group) of $\mathcal{G}$ by the group of 1-cocycles (respectively, the 1-coboundaries).

math.OA↗

arXiv2025

Uniqueness theorems for $L^p$-operator graph algebras

We continue the study of $L^p$-operator algebras associated with directed graphs initiated by Cortiñas and Rodríguez, and we establish $L^p$-analogs of both the gauge-invariant and the Cuntz-Krieger uniqueness theorems. The first of these asserts that for a graph $Q$, a gauge-equivariant spatial representation of its Leavitt path algebra $L_Q$ on an $L^p$-space generates an injective representation whenever the idempotents associated to the vertices of $Q$ are nonzero. The second of these theorems states that, in the setting just described, the same conclusion holds if gauge-equivariance is replaced by the assumption that every cycle in $Q$ has an entry. Additionally, we show that for acyclic graphs, such representations are automatically isometric. While our general approach is inspired by the proofs in the C*-algebra setting, a careful analysis of spatial representations of graphs on $L^p$-spaces is required. In particular, we exploit the interplay between analytical properties of Banach algebras, such as the role of hermitian elements, and geometric notions specific to $L^p$-spaces, such as spatial implementation.

math.FA↗

arXiv2025

Asymptotic lifting for completely positive maps

Let $A$ and $B$ be $C^*$-algebras with $A$ separable, let $I$ be an ideal in $B$, and let $ψ\colon A\to B/I$ be a completely positive contractive linear map. We show that there is a continuous family $Θ_t\colon A\to B$, for $t\in [1,\infty)$, of lifts of $ψ$ that are asymptotically linear, asymptotically completely positive and asymptotically contractive. If $ψ$ is of order zero, then $Θ_t$ can be chosen to have this property asymptotically. If $A$ and $B$ carry continuous actions of a second countable locally compact group $G$ such that $I$ is $G$-invariant and $ψ$ is equivariant, we show that the family $Θ_t$ can be chosen to be asymptotically equivariant. If a linear completely positive lift for $ψ$ exists, we can arrange that $Θ_t$ is linear and completely positive for all $t\in [1,\infty)$. In the equivariant setting, if $A$, $B$ and $ψ$ are unital, we show that asymptotically linear unital lifts are only guaranteed to exist if $G$ is amenable. This leads to a new characterization of amenability in terms of the existence of asymptotically equivariant unital sections for quotient maps.

math.OA↗

arXiv2025

Rigidity of pseudofunction algebras of ample groupoids

We show that a Hausdorff, ample groupoid $\mathcal{G}$ can be completely recovered from the $I$-norm completion of $C_c(\mathcal{G})$. More generally, we show that this is also the case for the algebra of symmetrized $p$-pseudofunctions, as well as for the reduced groupoid $L^p$-operator algebra, for $p\neq 2$. Our proofs are based on a new construction of an inverse semigroup built from Moore-Penrose invertible partial isometries in an $L^p$-operator algebra. Along the way, we verify a conjecture of Rakočević concerning the continuity of the Moore-Penrose inverse for $L^p$-operator algebras.

math.OA↗

arXiv2025

Fully noncentral Lie ideals and invariant additive subgroups in rings

We prove conditions ensuring that a Lie ideal or an invariant additive subgroup in a ring contains all additive commutators. A crucial assumption is that the subgroup is fully noncentral, that is, its image in every quotient is noncentral. For a unital algebra over a field of characteristic $\neq 2$ where every additive commutator is a sum of square-zero elements, we show that a fully noncentral subspace is a Lie ideal if and only if it is invariant under all inner automorphisms. This applies in particular to zero-product balanced algebras.

math.RA↗

arXiv2025

Simplicity of $C^*$-algebras of contracting self-similar groups

We show that the $C^*$-algebra associated by Nekrashevych to a contracting self-similar group is simple if and only if the corresponding complex $\ast$-algebra is simple. We also improve on Steinberg and Szakać's algorithm to determine if the $\ast$-algebra is simple. This provides an interesting class of non-Hausdorff amenable, effective and minimal ample groupoids for which simplicity of the $C^*$-algebra and the complex $\ast$-algebra are equivalent.

math.OA↗

arXiv2024

Semiprime ideals in C*-algebras

We show that a not necessarily closed ideal in a C*-algebra is semiprime if and only if it is idempotent, if and only if it is closed under square roots of positive elements. Among other things, it follows that prime and semiprime ideals in C*-algebras are automatically self-adjoint. To prove the above, we isolate and study a particular class of ideals, which we call Dixmier ideals. As it turns out, there is a rich theory of powers and roots for Dixmier ideals. We show that every ideal in a C*-algebra is squeezed by Dixmier ideals from inside and outside tightly in a suitable sense, from which we are able to deduce information about the ideal in the middle.

math.OA↗

arXiv2024

Actions on classifiable C*-algebras without equivariant property (SI)

We exhibit examples of actions of countable discrete groups on both simple and non-simple nuclear stably finite C*-algebras that are tracially amenable but not amenable. We furthermore obtain that, under the additional assumption of strict comparison, amenability is equivalent to tracial amenability plus the equivariant analogue of Matui--Sato's property (SI). By virtue of this equivalence, our construction yields the first known examples of actions on classifiable C*-algebras that do not have equivariant property (SI). We moreover show that such actions can be chosen to absorb the trivial action on the universal UHF algebra, thus showing that equivariant $\mathcal{Z}$-stability does not in general imply equivariant property (SI).

math.OA↗

arXiv2024

Rigidity results for $L^p$-operator algebras and applications

For $p\in [1,\infty)$, we show that every unital $L^p$-operator algebra contains a unique maximal $C^*$-subalgebra, which is always abelian if $p\neq 2$. Using this, we canonically associate to every unital $L^p$-operator algebra $A$ an étale groupoid $\mathcal{G}_A$, which in many cases of interest is a complete invariant for $A$. By identifying this groupoid for large classes of examples, we obtain a number of rigidity results that display a stark contrast with the case $p=2$; the most striking one being that of crossed products by topologically free actions. Our rigidity results give answers to questions concerning the existence of isomorphisms between different algebras. Among others, we show that for the $L^p$-analog $\mathcal{O}_2^p$ of the Cuntz algebra, there is no isometric isomorphism between $\mathcal{O}_2^p$ and $\mathcal{O}_2^p\otimes^p\mathcal{O}_2^p$, when $p\neq 2$. In particular, we deduce that there is no $L^p$-version of Kirchberg's absorption theorem, and that there is no $K$-theoretic classification of purely infinite simple amenable $L^p$-operator algebras for $p\neq 2$. Our methods also allow us to recover a folklore fact in the case of C*-algebras ($p=2$), namely that no isomorphism $\mathcal{O}_2^p\cong \mathcal{O}_2^p\otimes\mathcal{O}_2^p$ preserves the canonical Cartan subalgebras.

math.OA↗

arXiv2024

Essential freeness, allostery and $\mathcal{Z}$-stability of crossed products

We explore classifiability of crossed products of actions of countable amenable groups on compact, metrizable spaces. It is completely understood when such crossed products are simple, separable, unital, nuclear and satisfy the UCT: these properties are equivalent to the combination of minimality and topological freeness, and the challenge in this context is establishing $\mathcal{Z}$-stability. While most of the existing results in this direction assume freeness of the action, there exist numerous natural examples of minimal, topologically free (but not free) actions whose crossed products are classifiable. In this work, we take the first steps towards a systematic study of $\mathcal{Z}$-stability for crossed products beyond the free case, extending the available machinery around the small boundary property and almost finiteness to a more general setting. Among others, for actions of groups of polynomial growth with the small boundary property, we show that minimality and topological freeness are not just necessary, but also \emph{sufficient} conditions for classifiability of the crossed product. Our most general results apply to actions that are essentially free, a property weaker than freeness but stronger than topological freeness in the minimal setting. Very recently, M. Joseph produced the first examples of minimal actions of amenable groups which are topologically free and not essentially free. While the current machinery does not give any information for his examples, we develop ad-hoc methods to show that his actions have classifiable crossed products.

math.OA↗

arXiv2024

Products of commutators in matrix rings

Let $R$ be a ring and let $n\ge 2$. We discuss the question of whether every element in the matrix ring $M_n(R)$ is a product of (additive) commutators $[x,y]=xy-yx$, for $x,y\in M_n(R)$. An example showing that this does not always hold, even when $R$ is commutative, is provided. If, however, $R$ has Bass stable rank one, then under various additional conditions every element in $M_n(R)$ is a product of three commutators. Further, if $R$ is a division ring with infinite center, then every element in $M_n(R)$ is a product of two commutators. If $R$ is a field and $a\in M_n(R)$, then every element in $M_n(R)$ is a sum of elements of the form $[a,x][a,y]$ with $x,y\in M_n(R)$ if and only if the degree of the minimal polynomial of $a$ is greater than $2$.

math.RA↗

arXiv2024

Generalisations of Thompson's group V arising from purely infinite groupoids

We study a class of generalisations of Thompson's group $V$ arising naturally as topological full groups of purely infinite, minimal groupoids. In the process, we show that the derived subgroup of such a group is 2-generated whenever it is finitely generated and has no proper characters in full generality. We characterise this class of groupoids through a number of group-theoretic conditions on their full groups including vigor, the existence of suitable embeddings of $V$, and compressibility. We moreover give a complete abstract characterisation of those groups that arise as either topological full groups or derived subgroups of purely infinite, minimal groupoids. As an application, we describe all proper characters of the Brin-Higman-Thompson groups.

math.GR↗

arXiv2024

Rings and C*-algebras generated by commutators

We show that a unital ring is generated by its commutators as an ideal if and only if there exists a natural number $N$ such that every element is a sum of $N$ products of pairs of commutators. We show that one can take $N \leq 2$ for matrix rings, and that one may choose $N \leq 3$ for rings that contain a direct sum of matrix rings -- this in particular applies to C*-algebras that are properly infinite or have real rank zero. For Jiang-Su-stable C*-algebras, we show that $N\leq 6$ can be arranged. For arbitrary rings, we show that every element in the commutator ideal admits a power that is a sum of products of commutators. We prove that a C*-algebra cannot be a radical extension over a proper ideal, and we use this to deduce that a C*-algebra is generated by its commutators as a not necessarily closed ideal if and only if every element is a finite sum of products of pairs of commutators.

math.RA↗