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| 2. |
- Choi, Yemon, et al.
(author)
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Rigidity results for L p -operator algebras and applications
- 2024
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In: Advances in Mathematics. - 1090-2082 .- 0001-8708. ; 452
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Journal article (peer-reviewed)abstract
- For p∈[1,∞), we show that every unital Lp-operator algebra contains a unique maximal C⁎-subalgebra, which is always abelian if p≠2. Using this, we canonically associate to every unital Lp-operator algebra A an étale groupoid GA, 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 Lp-analog O2p of the Cuntz algebra, there is no isometric isomorphism between O2p and O2p⊗pO2p, when p≠2. In particular, we deduce that there is no Lp-version of Kirchberg's absorption theorem, and that there is no K-theoretic classification of purely infinite simple amenable Lp-operator algebras for p≠2. Our methods also allow us to recover a folklore fact in the case of C*-algebras (p=2), namely that no isomorphism O2≅O2⊗O2 preserves the canonical Cartan subalgebras.
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| 3. |
- Gardella, Eusebio, 1988, et al.
(author)
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Classifiability of crossed products by nonamenable groups
- 2023
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In: Journal für die Reine und Angewandte Mathematik. - : Walter de Gruyter GmbH. - 1435-5345 .- 0075-4102. ; 2023:797, s. 285-312
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Journal article (peer-reviewed)abstract
- 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, A2-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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| 4. |
- Gardella, Eusebio, 1988, et al.
(author)
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Dynamical comparison and Z-stability for crossed products of simple C ⁎ -algebras
- 2024
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In: Advances in Mathematics. - 1090-2082 .- 0001-8708. ; 438
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Journal article (peer-reviewed)abstract
- We establish Z-stability for crossed products of outer actions of amenable groups on 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 ∂eT(A), and such that the induced action G↷∂eT(A) is free. If G=Zd and the action G↷∂eT(A) is free and minimal, then we obtain McDuffness with respect to invariant traces, and thus Z-stability of the corresponding crossed product, also in case ∂eT(A) has infinite covering dimension.
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| 5. |
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| 6. |
- Gardella, Eusebio, et al.
(author)
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ISOMORPHISMS OF ALGEBRAS OF CONVOLUTION OPERATORS
- 2022
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In: Annales Scientifiques De L Ecole Normale Superieure. - : Societe Mathematique de France. - 0012-9593 .- 1873-2151. ; 55:5, s. 1432-1471
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Journal article (peer-reviewed)abstract
- For p, q E [1, oo) , 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 CVp(G) of p-convolvers and PMp(G) of p-pseudomeasures, for p # 2. More generally, we show that if CVp(G) is isometrically isomorphic to CVq(H) , with p, q # 2 , then G must be isomorphic to H and p and q are either equal or conjugate. This implies that there is no Lp-version of Connes' uniqueness of the hyperfinite II1-factor. Similar results apply to the algebra PFp(G) of p-pseudofunctions, gen-eralizing a classical result of Wendel. We also show that other Lp-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 PFp(G) and CVp(G): if any such algebra is reflexive and amenable, then G is finite.
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| 7. |
- Gardella, Eusebio, 1988, et al.
(author)
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PRIME IDEALS IN C*-ALGEBRAS AND APPLICATIONS TO LIE THEORY
- 2024
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In: Proceedings of the American Mathematical Society. - 0002-9939 .- 1088-6826. ; In Press
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Journal article (peer-reviewed)abstract
- . We show that every proper, dense ideal in a C & lowast;-algebra is contained in a prime ideal. It follows that a subset generates a C & lowast;-algebra as a not necessarily closed ideal if and only if it is not contained in any prime ideal. This allows us to transfer Lie theory results from prime rings to C & lowast;-algebras. For example, if a C & lowast;-algebra A is generated by its commutator subspace [A, A] as a ring, then [[A, A], [A, A]] = [A, A]. Further, given Lie ideals K and L in A, then [K, L] generates A as a not necessarily closed ideal if and only if [K, K] and [L, L] do, and moreover this implies that [K, L] = [A, A]. We also discover new properties of the subspace generated by square-zero elements and relate it to the commutator subspace of a C & lowast;-algebra.
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| 8. |
- Gardella, Eusebio, et al.
(author)
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Strongly outer actions of amenable groups on Z-stable nuclear C*-algebras
- 2022
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In: Journal des Mathématiques Pures et Appliquées. - : Elsevier BV. - 0021-7824. ; 162, s. 76-123
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Journal article (peer-reviewed)abstract
- Let A be a separable, unital, simple, Z-stable, nuclear C*-algebra, and let alpha: G -+ 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 the following are equivalent: (1) alpha is strongly outer; (2) alpha (R) idZ has the weak tracial Rokhlin property. If G is moreover residually finite, the above conditions are also equivalent to (3) alpha (R) idZ has finite Rokhlin dimension (in fact, at most 2). If partial differential eT(A) is furthermore compact, has finite covering dimension, and the orbit space partial differential eT(A)/G is Hausdorff, we generalize results by Matui and Sato to show that alpha is cocycle conjugate to alpha (R) idZ, even if alpha is not strongly outer. In particular, in this case the equivalences above hold for alpha in place of alpha (R) idZ. In the course of the proof, we develop equivariant versions of complemented partitions of unity and uniform property Gamma as technical tools of independent interest. (c) 2022 The Author(s). Published by Elsevier Masson SAS. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
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| 9. |
- Gardella, Eusebio, 1988, et al.
(author)
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Tracially amenable actions and purely infinite crossed products
- 2024
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In: Mathematische Annalen. - 0025-5831 .- 1432-1807. ; In Press
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Journal article (peer-reviewed)abstract
- We introduce the notion of tracial amenability for actions of discrete groups on unital, tracial C∗-algebras, as a weakening of amenability where all the relevant approximations are done in the uniform trace norm. We characterize tracial amenability with various equivalent conditions, including topological amenability of the induced action on the trace space. Our main result concerns the structure of crossed products: for groups containing the free group F2, we show that outer, tracially amenable actions on simple, unital, Z-stable C∗-algebras always have purely infinite crossed products. Finally, we give concrete examples of tracially amenable actions of free groups on simple, unital AF-algebras.
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| 10. |
- Gardella, Eusebio, et al.
(author)
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Tracially amenable actions and purely infinite crossed products
- 2024
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In: MATHEMATISCHE ANNALEN. - 0025-5831 .- 1432-1807.
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Journal article (peer-reviewed)abstract
- We introduce the notion of tracial amenability for actions of discrete groups on unital, tracial C * \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$<^>*$$\end{document} -algebras, as a weakening of amenability where all the relevant approximations are done in the uniform trace norm. We characterize tracial amenability with various equivalent conditions, including topological amenability of the induced action on the trace space. Our main result concerns the structure of crossed products: for groups containing the free group F 2 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_2$$\end{document} , we show that outer, tracially amenable actions on simple, unital, Z \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {Z}$$\end{document} -stable C * \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$<^>*$$\end{document} -algebras always have purely infinite crossed products. Finally, we give concrete examples of tracially amenable actions of free groups on simple, unital AF-algebras.
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| 11. |
- Gardella, Eusebio, et al.
(author)
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Zero-product balanced algebras
- 2023
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In: Linear Algebra and Its Applications. - 0024-3795. ; 670, s. 121-153
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Journal article (peer-reviewed)abstract
- We say that an algebra is zero-product balanced if ab 0 c and a 0 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 introduced and developed by Bresar, Villena and others. 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 commutative, reduced 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. (c) 2023 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/).
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