Large scale geometry of Banach-Lie groups

Kód výsledku v 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>

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

Large scale geometry of Banach-Lie groups

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

Large scale geometry of Banach-Lie groups

Popis výsledku anglicky

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.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GJ19-05271Y" target="_blank" >GJ19-05271Y: Grupy a jejich akce, operátorové algebry a deskriptivní teorie množin</a><br>

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2022

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

American Mathematical Society. Transactions

ISSN

0002-9947

e-ISSN

1088-6850

Svazek periodika

375

Číslo periodika v rámci svazku

4

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

55

Strana od-do

2827-2881

Kód UT WoS článku

000768789700018

EID výsledku v databázi Scopus

2-s2.0-85126467891