Large scale geometry of Banach-Lie groups

0555458 - MÚ 2023 RIV US eng J - Journal Article
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
R&D Projects: GA ČR(CZ) GJ19-05271Y
Institutional support: RVO:67985840
Keywords : Banach-Lie groups * large scale geometry * unitary groups * Haagerup property
OECD category: Pure mathematics
Impact factor: 1.3, year: 2022
Method of publishing: 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.
Permanent Link: http://hdl.handle.net/11104/0329974