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Large scale geometry of Banach-Lie groups
- 1.0555458 - MÚ 2023 RIV US eng J - Článek v odborném periodiku
Ando, H. - Doucha, Michal - Matsuzawa, Y.
Large scale geometry of Banach-Lie groups.
American Mathematical Society. Transactions. Roč. 375, č. 4 (2022), s. 2827-2881. ISSN 0002-9947. E-ISSN 1088-6850
Grant CEP: GA ČR(CZ) GJ19-05271Y
Institucionální podpora: RVO:67985840
Klíčová slova: Banach-Lie groups * large scale geometry * unitary groups * Haagerup property
Obor OECD: Pure mathematics
Impakt faktor: 1.3, rok: 2022
Způsob publikování: Open access
https://doi.org/10.1090/tran/8576
We initiate the large scale geometric study of Banach-Lie groups, especially of linear Banach-Lie groups. We show that the exponential length, originally introduced by Ringrose for unitary groups of -algebras, defines the quasi-isometry type of any connected Banach-Lie group. As an illustrative example, we consider unitary groups of separable abelian unital -algebras with spectrum having finitely many components, which we classify up to topological isomorphism and up to quasi-isometry, in order to highlight the difference. The main results then concern the Haagerup property, and Properties (T) and (FH). We present the first non-trivial non-abelian and non-locally compact groups having the Haagerup property, most of them being non-amenable. These are the groups , where is a semifinite von Neumann algebra with a normal faithful semifinite trace . Finally, we investigate the groups , which are closed subgroups of generated by elementary matrices, where is a unital Banach algebra. We show that for, all these groups have Property (T) and they are unbounded, so they have Property (FH) non-trivially. On the other hand, if is an infinite-dimensional unital -algebra, then does not have the Haagerup property. If is moreover abelian and separable, then does not have the Haagerup property.
Trvalý link: http://hdl.handle.net/11104/0329974
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