| 1. |
- 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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| 2. |
- 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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| 3. |
- 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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