Large scale geometry of Banach-Lie groups

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F22%3A00555458" target="_blank" >RIV/67985840:_____/22:00555458 - isvavai.cz</a>
Result on the web
<a href="https://doi.org/10.1090/tran/8576" target="_blank" >https://doi.org/10.1090/tran/8576</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1090/tran/8576" target="_blank" >10.1090/tran/8576</a>

Result language
angličtina
Original language name
Large scale geometry of Banach-Lie groups
Original language description
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.
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics

Project
<a href="/en/project/GJ19-05271Y" target="_blank" >GJ19-05271Y: Groups and their actions, operator algebras, and descriptive set theory</a><br>
Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2022
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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