Lie theoretic approach to unitary groups of C*-algebras

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F25%3A00605688" target="_blank" >RIV/67985840:_____/25:00605688 - isvavai.cz</a>

Výsledek na webu

<a href="https://doi.org/10.1090/tran/9330" target="_blank" >https://doi.org/10.1090/tran/9330</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1090/tran/9330" target="_blank" >10.1090/tran/9330</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Lie theoretic approach to unitary groups of C*-algebras

Popis výsledku v původním jazyce

Following Robert’s [J. Reine Angew. Math. 756 (2019), pp. 285–319], we study the structure of unitary groups and groups of approximately inner automorphisms of unital C∗-algebras, taking advantage of the former being Banach-Lie groups. For a given unital C∗-algebra A, we provide a description of the closed normal subgroup structure of the connected component of the identity of the unitary group, denoted by UA, resp. of the subgroup of approximately inner automorphisms induced by the connected component of the identity of the unitary group, denoted by VA, in terms of perfect ideals, i.e. ideals admitting no characters. When the unital algebra is locally AF, we show that there is a one-to-one correspondence between closed normal subgroups of VA and perfect ideals of the algebra, which can be in the separable case conveniently described using Bratteli diagrams, in particular showing that every closed normal subgroup of VA is perfect. We also characterize unital C∗algebras A such that UA, resp. VA are topologically simple, generalizing the main results of Robert [J. Reine Angew. Math. 756 (2019), pp. 285–319] from [26]. In the other way round, under certain conditions, we characterize simplicity of the algebra in terms of the structure of the unitary group. This in particular applies to reduced group C∗-algebras of discrete groups and we show that when A is a reduced group C∗-algebra of a non-amenable countable discrete group, then A is simple if and only if UA/T is topologically simple.

Název v anglickém jazyce

Lie theoretic approach to unitary groups of C*-algebras

Popis výsledku anglicky

Following Robert’s [J. Reine Angew. Math. 756 (2019), pp. 285–319], we study the structure of unitary groups and groups of approximately inner automorphisms of unital C∗-algebras, taking advantage of the former being Banach-Lie groups. For a given unital C∗-algebra A, we provide a description of the closed normal subgroup structure of the connected component of the identity of the unitary group, denoted by UA, resp. of the subgroup of approximately inner automorphisms induced by the connected component of the identity of the unitary group, denoted by VA, in terms of perfect ideals, i.e. ideals admitting no characters. When the unital algebra is locally AF, we show that there is a one-to-one correspondence between closed normal subgroups of VA and perfect ideals of the algebra, which can be in the separable case conveniently described using Bratteli diagrams, in particular showing that every closed normal subgroup of VA is perfect. We also characterize unital C∗algebras A such that UA, resp. VA are topologically simple, generalizing the main results of Robert [J. Reine Angew. Math. 756 (2019), pp. 285–319] from [26]. In the other way round, under certain conditions, we characterize simplicity of the algebra in terms of the structure of the unitary group. This in particular applies to reduced group C∗-algebras of discrete groups and we show that when A is a reduced group C∗-algebra of a non-amenable countable discrete group, then A is simple if and only if UA/T is topologically simple.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GF22-07833K" target="_blank" >GF22-07833K: Homogenita a generičnost a metrických struktur - grup, dynamických systémů, Banachových prostorů a C*-algeber</a><br>

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2025

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

378

Číslo periodika v rámci svazku

3

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

24

Strana od-do

2007-2030

Kód UT WoS článku

001385882400001

EID výsledku v databázi Scopus

2-s2.0-85217147931