Lie theoretic approach to unitary groups of C*-algebras

Result code in 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>

Result on the web

<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>

Result language

angličtina

Original language name

Lie theoretic approach to unitary groups of C*-algebras

Original language description

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.

Czech name

—

Czech description

—

Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

—

OECD FORD branch

10101 - Pure mathematics

Project

<a href="/en/project/GF22-07833K" target="_blank" >GF22-07833K: Homogeneity and Genericity of Metric Structures - Groups, Dynamical Systems, Banach Spaces and C*-Algebras</a><br>

Continuities

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year

2025

Confidentiality

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

Name of the periodical

American Mathematical Society. Transactions

ISSN

0002-9947

e-ISSN

1088-6850

Volume of the periodical

378

Issue of the periodical within the volume

3

Country of publishing house

US - UNITED STATES

Number of pages

24

Pages from-to

2007-2030

UT code for WoS article

001385882400001

EID of the result in the Scopus database

2-s2.0-85217147931