Closed ideals of operators on the Tsirelson and Schreier spaces

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F20%3A00525128" target="_blank" >RIV/67985840:_____/20:00525128 - isvavai.cz</a>

Výsledek na webu

<a href="https://doi.org/10.1016/j.jfa.2020.108668" target="_blank" >https://doi.org/10.1016/j.jfa.2020.108668</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1016/j.jfa.2020.108668" target="_blank" >10.1016/j.jfa.2020.108668</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Closed ideals of operators on the Tsirelson and Schreier spaces

Popis výsledku v původním jazyce

Let B(X) denote the Banach algebra of bounded operators on X, where X is either Tsirelson's Banach space or the Schreier space of order n for some n∈N. We show that the lattice of closed ideals of B(X) has a very rich structure, in particular B(X) contains at least continuum many maximal ideals. Our approach is to study the closed ideals generated by the basis projections. Indeed, the unit vector basis is an unconditional basis for each of the above spaces, so there is a basis projection PN∈B(X) corresponding to each non-empty subset N of N. A closed ideal of B(X) is spatial if it is generated by PN for some N. We can now state our main conclusions as follows: • the family of spatial ideals lying strictly between the ideal of compact operators and B(X) is non-empty and has no minimal or maximal elements, • for each pair I⫋J of spatial ideals, there is a family {ΓL:L∈Δ}, where the index set Δ has the cardinality of the continuum, such that ΓL is an uncountable chain of spatial ideals, ⋃ΓL is a closed ideal that is not spatial, and I⫋L⫋JandL+M‾=J whenever L,M∈Δ are distinct and L∈ΓL, M∈ΓM.

Název v anglickém jazyce

Closed ideals of operators on the Tsirelson and Schreier spaces

Popis výsledku anglicky

Let B(X) denote the Banach algebra of bounded operators on X, where X is either Tsirelson's Banach space or the Schreier space of order n for some n∈N. We show that the lattice of closed ideals of B(X) has a very rich structure, in particular B(X) contains at least continuum many maximal ideals. Our approach is to study the closed ideals generated by the basis projections. Indeed, the unit vector basis is an unconditional basis for each of the above spaces, so there is a basis projection PN∈B(X) corresponding to each non-empty subset N of N. A closed ideal of B(X) is spatial if it is generated by PN for some N. We can now state our main conclusions as follows: • the family of spatial ideals lying strictly between the ideal of compact operators and B(X) is non-empty and has no minimal or maximal elements, • for each pair I⫋J of spatial ideals, there is a family {ΓL:L∈Δ}, where the index set Δ has the cardinality of the continuum, such that ΓL is an uncountable chain of spatial ideals, ⋃ΓL is a closed ideal that is not spatial, and I⫋L⫋JandL+M‾=J whenever L,M∈Δ are distinct and L∈ΓL, M∈ΓM.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GJ19-07129Y" target="_blank" >GJ19-07129Y: Metody lineární analýzy v operátorových algebrách a naopak</a><br>

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2020

Kód důvěrnosti údajů

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

Název periodika

Journal of Functional Analysis

ISSN

0022-1236

e-ISSN

—

Svazek periodika

279

Číslo periodika v rámci svazku

8

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

28

Strana od-do

108668

Kód UT WoS článku

000560373600006

EID výsledku v databázi Scopus

2-s2.0-85086106962