The order topology on duals of C*-algebras and von Neumann algebras

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21230%2F20%3A00346256" target="_blank" >RIV/68407700:21230/20:00346256 - isvavai.cz</a>

Výsledek na webu

<a href="https://doi.org/10.4064/sm190108-11-7" target="_blank" >https://doi.org/10.4064/sm190108-11-7</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.4064/sm190108-11-7" target="_blank" >10.4064/sm190108-11-7</a>

Jazyk výsledku

angličtina

Název v původním jazyce

The order topology on duals of C*-algebras and von Neumann algebras

Popis výsledku v původním jazyce

For a von Neumann algebra M, we study the order topology associated to the hermitian part M-*(s), and to intervals of the predual M-*. It is shown that the order topology on M-*(s) coincides with the topology induced by the norm. In contrast, it is proved that the condition of having the order topology, associated to the interval [0, phi], equal to the topology induced by the norm, for every phi is an element of M-*(+), is necessary and sufficient for the commutativity of M. It is also proved that if phi is a positive bounded functional on a C*-algebra A, then the norm-null sequences in [0, phi] coincide with the null sequences, with respect to the order topology on [0, phi], if and only if the von Neumann algebra pi(phi)(A)' is of finite type (where pi(phi) denotes the corresponding GNS representation). This fact allows us to give a new topological characterization of finite von Neumann algebras. Moreover, we demonstrate that convergence to zero for norm and order topology, on order-bounded parts of dual spaces, are inequivalent for all C*-algebras that are not of type I.

Název v anglickém jazyce

The order topology on duals of C*-algebras and von Neumann algebras

Popis výsledku anglicky

For a von Neumann algebra M, we study the order topology associated to the hermitian part M-*(s), and to intervals of the predual M-*. It is shown that the order topology on M-*(s) coincides with the topology induced by the norm. In contrast, it is proved that the condition of having the order topology, associated to the interval [0, phi], equal to the topology induced by the norm, for every phi is an element of M-*(+), is necessary and sufficient for the commutativity of M. It is also proved that if phi is a positive bounded functional on a C*-algebra A, then the norm-null sequences in [0, phi] coincide with the null sequences, with respect to the order topology on [0, phi], if and only if the von Neumann algebra pi(phi)(A)' is of finite type (where pi(phi) denotes the corresponding GNS representation). This fact allows us to give a new topological characterization of finite von Neumann algebras. Moreover, we demonstrate that convergence to zero for norm and order topology, on order-bounded parts of dual spaces, are inequivalent for all C*-algebras that are not of type I.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

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OECD FORD obor

10101 - Pure mathematics

Projekt

Výsledek vznikl pri realizaci vícero projektů. Více informací v záložce Projekty.

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2020

Kód důvěrnosti údajů

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

Název periodika

Studia Mathematica

ISSN

0039-3223

e-ISSN

1730-6337

Svazek periodika

254

Číslo periodika v rámci svazku

3

Stát vydavatele periodika

PL - Polská republika

Počet stran výsledku

18

Strana od-do

219-236

Kód UT WoS článku

000558107800001

EID výsledku v databázi Scopus

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