arXiv subjects

Eusebio Gardella

Publications and source records attributed to Eusebio Gardella.

At least 37 records · Page 2Linked to original sources

arXiv2023

Zero-product balanced algebras

We say that an algebra is zero-product balanced if $ab\otimes c$ and $a\otimes bc$ agree modulo tensors of elements with zero-product. This is closely related to but more general than the notion of a zero-product determined algebra of Brešar, Grašič and Ortega. Every surjective, zero-product preserving map from a zero-product balanced algebra is automatically a weighted epimorphism, and this implies that zero-product balanced algebras are determined by their linear and zero-product structure. Further, the commutator subspace of a zero-product balanced algebra can be described in terms of square-zero elements. We show that a semiprime, commutative algebra is zero-product balanced if and only if it is generated by idempotents. It follows that every commutative, zero-product balanced algebra is spanned by nilpotent and idempotent elements. We deduce a dichotomy for unital, zero-product balanced algebras: They either admit a character or are generated by nilpotents.

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arXiv2023

Classifiability of crossed products by nonamenable groups

We show that all amenable, minimal actions of a large class of nonamenable countable groups on compact metric spaces have dynamical comparison. This class includes all nonamenable hyperbolic groups, many HNN-extensions, nonamenable Baumslag-Solitar groups, a large class of amalgamated free products, lattices in many Lie groups, $\widetilde{A}_2$-groups, as well as direct products of the above with arbitrary countable groups. As a consequence, crossed products by amenable, minimal and topologically free actions of such groups on compact metric spaces are Kirchberg algebras in the UCT class, and are therefore classified by $K$-theory.

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arXiv2022

The modern theory of Cuntz semigroups of C*-algebras

We give a detailed introduction to the theory of Cuntz semigroups for C*-algebras. Beginning with the most basic definitions and technical lemmas, we present several results of historical importance, such as Cuntz's theorem on the existence of quasitraces, Rørdam's proof that $\mathcal{Z}$-stability implies strict comparison, and Toms' example of a non $\mathcal{Z}$-stable simple, nuclear C*-algebra. We also give the reader an extensive overview of the state of the art and the modern approach to the theory, including the recent results for C*-algebras of stable rank one (for example, the Blackadar-Handelman conjecture and the realization of ranks), as well as the abstract study of the Cuntz category $\mathbf{Cu}$.

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arXiv2022

Dynamical comparison and $\mathcal{Z}$-stability for crossed products of simple $C^*$-algebras

We establish $\mathcal{Z}$-stability for crossed products of outer actions of amenable groups on $\mathcal{Z}$-stable $C^*$-algebras under a mild technical assumption which we call McDuff property with respect to invariant traces. We obtain such result using a weak form of dynamical comparison, which we verify in great generality. We complement our results by proving that McDuffness with respect to invariant traces is automatic in many cases of interest. This is the case, for instance, for every action of an amenable group $G$ on a classifiable $C^*$-algebra $A$ whose trace space $T(A)$ is a Bauer simplex with finite dimensional boundary $\partial_e T(A)$, and such that the induced action $G\curvearrowright \partial_eT(A)$ is free. If $G = \mathbb{Z}^d$ and the action $G\curvearrowright \partial_eT(A)$ is free and minimal, then we obtain McDuffness with respect to invariant traces, and thus $\mathcal{Z}$-stability of the corresponding crossed product, also in case $\partial_e T(A)$ has infinite covering dimension.

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arXiv2022

Weighted homomorphisms between C*-algebras

We show that a bounded, linear map between C*-algebras is a weighted $\ast$-homomorphism (the central compression of a $\ast$-homomorphism) if and only if it preserves zero-products, range-orthogonality, and domain-orthogonality. It follows that a self-adjoint, bounded, linear map is a weighted $\ast$-homomorphism if and only if it preserves zero-products. As an application we show that a linear map between C*-algebras is completely positive, order zero in the sense of Winter-Zacharias if and only if it is positive and preserves zero-products.

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arXiv2022

Decomposable partial actions

We define the decomposition property for partial actions of discrete groups on $C^*$-algebras. Decomposable partial systems appear naturally in practice, and many commonly occurring partial actions can be decomposed into partial actions with the decomposition property. For instance, any partial action of a finite group is an iterated extension of decomposable systems. Partial actions with the decomposition property are always globalizable and amenable, regardless of the acting group, and their globalization can be explicitly described in terms of certain global sub-systems. A direct computation of their crossed products is also carried out. We show that partial actions with the decomposition property behave in many ways like global actions of finite groups (even when the acting group is infinite), which makes their study particularly accessible. For example, there exists a canonical faithful conditional expectation onto the fixed point algebra, which is moreover a corner in the crossed product in a natural way. (Both of these facts are in general false for partial actions of finite groups.) As an application, we show that freeness of a topological partial action with the decomposition property is equivalent to its fixed point algebra being Morita equivalent to its crossed product. We also show by example that this fails for general partial actions of finite groups.

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arXiv2022

Partial C*-dynamics and Rokhlin dimension

We develop the notion of Rokhlin dimension for partial actions of finite groups, extending the well-established theory for global systems. The partial setting exhibits phenomena that cannot be expected for global actions, usually stemming from the fact that virtually all averaging arguments for finite group actions completely break down for partial systems. For example, fixed point algebra and crossed product are not in general Morita equivalent, and there is in general no local approximation of the crossed product $A\rtimes G$ by matrices over $A$. By using decomposition arguments for partial actions of finite groups, we show that a number of structural properties are preserved by formation of crossed products, including finite stable rank, finite nuclear dimension, and absorption of a strongly self-absorbing $C^*$-algebra. For partial actions with the Rokhlin property, being an AF-algebra is also preserved. Some of our results are new even in the global case. We also study the Rokhlin dimension of globalizable actions: while in general it differs from the Rokhlin dimension of its globalization, we show that they agree if the coefficient algebra is unital. For topological partial actions on spaces of finite covering dimension, we show that finiteness of the Rokhlin dimension is equivalent to freeness, thus providing a large class of examples to which our theory applies.

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arXiv2021

Strongly outer actions of amenable groups on $\mathcal{Z}$-stable nuclear $C^*$-algebras

Let $A$ be a separable, unital, simple, $\mathcal{Z}$-stable, nuclear $C^*$-algebra, and let $α\colon G\to \mathrm{Aut}(A)$ be an action of a discrete, countable, amenable group. Suppose that the orbits of the action of $G$ on $T(A)$ are finite and that their cardinality is bounded. We show that $α$ is strongly outer if and only if $α\otimes\mathrm{id}_{\mathcal{Z}}$ has the weak tracial Rokhlin property. If $G$ is moreover residually finite, these conditions are also equivalent to $α\otimes\mathrm{id}_{\mathcal{Z}}$ having finite Rokhlin dimension (in fact, at most 2). If $\partial_eT(A)$ is furthermore compact, has finite covering dimension, and the orbit space $\partial_eT(A)/G$ is Hausdorff, we generalize results by Matui and Sato to show that $α$ is cocycle conjugate to $α\otimes\mathrm{id}_{\mathcal{Z}}$, even if $α$ is not strongly outer. In particular, in this case the equivalences above hold for $α$ in place of $α\otimes\mathrm{id}_{\mathcal{Z}}$. In the course of the proof, we develop equivariant versions of complemented partitions of unity and uniform property $Γ$ as technical tools of independent interest.

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arXiv2021

Equivariant KK-theory and the continuous Rokhlin property

We introduce and study the continuous Rokhlin property for actions of compact groups on C*-algebras. An important technical result is a characterization of the continuous Rokhlin property in terms of asymptotic retracts. As a consequence, we derive strong KK-theoretical obstructions to the continuous Rokhlin property. Using these, we show that the UCT is preserved under formation of crossed products and passage to fixed point algebras by such actions, even in the absence of nuclearity. Our analysis of the KK-theory of the crossed product allows us to prove a $\mathbb{T}$-equivariant version of Kirchberg-Phillips: two circle actions with the continuous Rokhlin property on Kirchberg algebras are conjugate whenever they are KK$^{\mathbb{T}}$-equivalent. In the presence of the UCT, this is equivalent to having isomorphic equivariant K-theory. We moreover characterize the KK$^{\mathbb{T}}$-theoretical invariants that arise in this way. Finally, we identify a KK$^{\mathbb{T}}$-theoretic obstruction to the continuous property, which is shown to be the only obstruction in the setting of Kirchberg algebras. We show by means of explicit examples that the Rokhlin property is strictly weaker than the continuous Rokhlin property.

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arXiv2021

Equivariant bundles and absorption

For a locally compact group $G$ and a strongly self-absorbing $G$-algebra $(\mathcal{D},δ)$, we obtain a new characterization of absorption of a strongly self-absorbing action using almost equivariant completely positive maps into the underlying algebra. The main technical tool to obtain this characterization is the existence of almost equivariant lifts for equivariant completely positive maps, proved in recent work of the authors. This characterization is then used to show that an equivariant $C_0(X)$-algebra with $\mathrm{dim}_{\mathrm{cov}}(X)<\infty$ is $(\mathcal{D},δ)$-stable if and only if all of its fibers are, extending a result of Hirshberg, Rørdam and Winter to the equivariant setting. The condition on the dimension of $X$ is known to be necessary, and we show that it can be removed if, for example, the bundle is locally trivial.

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arXiv2020

KK-theory of circle actions with the Rokhlin property

We investigate the structure of circle actions with the Rokhlin property, particularly in relation to equivariant $KK$-theory. Our main results are $\mathbb{T}$-equivariant versions of celebrated results of Kirchberg: any Rokhlin action on a separable, nuclear C*-algebra is $KK^\mathbb{T}$-equivalent to a Rokhlin action on a Kirchberg algebra; any Rokhlin action on an exact separable C*-algebra embeds equivariantly into $\mathcal{O}_2$ (with its unique Rokhlin action); and two circle actions with the Rokhlin property on a Kirchberg algebra are conjugate if and only if they are $KK^\mathbb{T}$-equivalent. In the presence of the UCT, $KK^\mathbb{T}$-equivalence for Rokhlin actions reduces to isomorphism of a $K$-theoretical invariant, namely of a canonical pure extension naturally associated to any Rokhlin action, and we provide a complete description of the extensions that arise from actions on nuclear C*-algebras. In contrast with the non-equivariant setting, an isomorphism between the $K^\mathbb{T}$-theories of Rokhlin actions on Kirchberg algebras does not necessarily lift to a $KK^\mathbb{T}$-equivalence.

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arXiv2020

Rokhlin dimension: duality, tracial properties, and crossed products

We study compact group actions with finite Rokhlin dimension, particularly in relation to crossed products. For example, we characterize the duals of such actions, generalizing previous partial results for the Rokhlin property. As an application, we determine the ideal structure of their crossed products. Under the assumption of so-called commuting towers, we show that taking crossed products by such actions preserves a number of relevant classes of C*-algebras, including: $D$-absorbing C*-algebras, where $D$ is a strongly self-absorbing C*-algebra, stable C*-algebras, C*-algebras with finite nuclear dimension (or decomposition rank), C*-algebras with finite stable rank (or real rank), and C*-algebras whose K-theory is either trivial, rational, or $n$-divisible for $n\in\mathbb{N}$. The combination of nuclearity and the UCT is also shown to be preserved by these actions. Some of these results are new even in the well-studied case of the Rokhlin property. Additionally, and under some technical assumptions, we show that finite Rokhlin dimension with commuting towers implies the (weak) tracial Rokhlin property. At the core of our arguments is a certain local approximation of the crossed product by a continuous $C(X)$-algebra with fibers that are stably isomorphic to the underlying algebra. The space $X$ is computed in some cases of interest, and we use its description to construct a $\mathbb{Z}_2$-action on a unital AF-algebra and on a unital Kirchberg algebra satisfying the UCT, whose Rokhlin dimensions with and without commuting towers are finite but do not agree.

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arXiv2020

Circle actions on UHF-absorbing $C^*$-algebras

We study circle actions with the Rokhlin property, in relation to their restrictions to finite subgroups. We construct examples showing the following: the restriction of a circle action with the Rokhlin property (even on a real rank zero $C^*$-algebra), need not have the Rokhlin property; and even if every restriction of a given circle action has the Rokhlin property, the circle action itself need not have it. As a positive result, we show that the restriction of a circle action with the Rokhlin property to the subgroup $\mathbb{Z}_n$ has the Rokhlin property if the underlying algebra absorbs $M_{n^\infty}$. The condition on the algebra is also necessary in most cases of interest. Despite the fact that there are no circle actions with the Rokhlin property on UHF-algebras, we construct many such actions on certain UHF-absorbing simple AT-algebras. Additionally, we show that circle actions with the Rokhlin property on $\mathcal{O}_2$-absorbing $C^*$-algebras are generic, in a suitable sense.

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arXiv2020

Preduals and complementation of spaces of bounded linear operators

For Banach spaces X and Y, we establish a natural bijection between preduals of Y and preduals of L(X,Y) that respect the right L(X)-module structure. If X is reflexive, it follows that there is a unique predual making L(X) into a dual Banach algebra. This removes the condition that X have the approximation property in a result of Daws. We further establish a natural bijection between projections that complement Y in its bidual and L(X)-linear projections that complement L(X,Y) in its bidual. It follows that Y is complemented in its bidual if and only if L(X,Y) is (either as a module or as a Banach space). Our results are new even in the well-studied case of isometric preduals.

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arXiv2020

Strongly outer actions of amenable groups on $\mathcal{Z}$-stable $C^*$-algebras

Let $A$ be a separable, unital, simple, $\mathcal{Z}$-stable, nuclear $C^*$-algebra, and let $α\colon G\to \mathrm{Aut}(A)$ be an action of a countable amenable group $G$. If the trace space $T(A)$ is a Bauer simplex and the action of $G$ on $\partial_eT(A)$ has finite orbits and Hausdorff orbit space, we show that $α$ is strongly outer if and only if $α\otimes\mathrm{id}_{\mathcal{Z}}$ has the weak tracial Rokhlin property. If $G$ is moreover residually finite, then these conditions are also equivalent to $α\otimes\mathrm{id}_{\mathcal{Z}}$ having finite Rokhlin dimension (in fact, at most 2). When the covering dimension of $\partial_eT(A)$ is finite, we prove that $α$ is cocycle conjugate to $α\otimes\mathrm{id}_{\mathcal{Z}}$. In particular, the equivalences above hold for $α$ in place of $α\otimes\mathrm{id}_{\mathcal{Z}}$.

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arXiv2019

A modern look at algebras of operators on $L^p$-spaces

The study of operator algebras on Hilbert spaces, and C*-algebras in particular, is one of the most active areas within Functional Analysis. A natural generalization of these is to replace Hilbert spaces (which are $L^2$-spaces) with $L^p$-spaces, for $p\in [1,\infty)$. The study of such algebras of operators is notoriously more challenging, due to the lack of orthogonality in $L^p$-spaces. We give a modern overview of a research area whose beginnings can be traced back to the 50's, and that has seen renewed attention in the last decade through the infusion of new techniques. The combination of these new ideas with old tools was the key to answer some long standing questions. Among others, we provide a description of all unital contractive homomorphisms between algebras of $p$-pseudofunctions of groups.

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arXiv2019

The local-triviality dimension of actions of compact quantum groups

We define the local-triviality dimension for actions of compact quantum groups on unital C*-algebras. The resulting compact quantum principal bundle is said to be locally trivial when this dimension is finite. For commutative C*-algebras, this notion recovers the standard definition of local triviality of compact principal bundles. We prove that actions with finite local-triviality dimension are automatically free. Then we apply this new notion to prove the noncommutative Borsuk-Ulam-type conjecture under the assumption that a compact quantum group admits a non-trivial classical subgroup whose induced action has finite local-triviality dimension. This is a noncommutative extension of the Borsuk-Ulam-type theorem for locally trivial principal bundles.

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arXiv2018

Isomorphisms of Algebras of Convolution Operators

For $p,q\in [1,\infty)$, we study the isomorphism problem for the $p$- and $q$-convolution algebras associated to locally compact groups. While it is well known that not every group can be recovered from its group von Neumann algebra, we show that this is the case for the algebras $\mathrm{CV}_p(G)$ of $p$-convolvers and $\mathrm{PM}_p(G)$ of $p$-pseudomeasures, for $p\neq 2$. More generally, we show that if $\mathrm{CV}_p(G)$ is isometrically isomorphic to $\mathrm{CV}_q(H)$, with $p,q\neq 2$, then $G$ must be isomorphic to $H$ and $p$ and $q$ are either equal or conjugate. This implies that there is no $L^p$-version of Connes' uniqueness of the hyperfinite II$_1$-factor. Similar results apply to the algebra $\mathrm{PF}_p(G)$ of $p$-pseudofunctions, generalizing a classical result of Wendel. We also show that other $L^p$-rigidity results for groups can be easily recovered and extended using our main theorem. Our results answer questions originally formulated in the work of Herz in the 70's. Moreover, our methods reveal new information about the Banach algebras in question. As a non-trivial application, we verify the reflexivity conjecture for all Banach algebras lying between $\mathrm{PF}_p(G)$ and $\mathrm{CV}_p(G)$: if any such algebra is reflexive and amenable, then $G$ is finite.

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