Hilbert C*-module independence

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21230%2F24%3A00381379" target="_blank" >RIV/68407700:21230/24:00381379 - isvavai.cz</a>

Výsledek na webu

<a href="https://doi.org/10.1002/mana.202200472" target="_blank" >https://doi.org/10.1002/mana.202200472</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1002/mana.202200472" target="_blank" >10.1002/mana.202200472</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Hilbert C*-module independence

Popis výsledku v původním jazyce

We introduce the notion of Hilbert C*-module independence: Let A be a unital C*-algebra and let E-i subset of E, i = 1, 2, be ternary subspaces of a Hilbert A-module E. Then, E-1 and E-2 are said to be Hilbert C*-module independent if there are positive constants m and M such that for every state phi(i) on < E-i, E-i >, = 1, 2, there exists a state phi on A such that m phi(i)(vertical bar x vertical bar) <= phi(vertical bar x vertical bar) <= M phi(i)(vertical bar x vertical bar(2))(1/2), for all x is an element of E-i,E- i = 1, 2. We show that it is a natural generalization of the notion of C*-independence of C*-algebras. Moreover, we demonstrate that even in the case of C*-algebras, this concept of independence is new and has a nice characterization in terms of Hahn-Banach-type extensions. We show that if < E-1, E-1 > has the quasi extension property and z is an element of E-1 boolean AND E-2 with vertical bar vertical bar z vertical bar vertical bar = 1, then vertical bar vertical bar z vertical bar vertical bar = 1. Several characterizations of Hilbert C*-module independence and a new characterization of C*-independence are given. One of characterizations states that if z(0) is an element of E-1 boolean AND E-2 is such that < z(0), z(0)> = 1, then E-1 and E-2 are Hilbert C*-module independent if and only if vertical bar vertical bar < x, z(0)> < y, z(0)> vertical bar vertical bar = vertical bar vertical bar < x, z(0)> vertical bar vertical bar vertical bar vertical bar < y, z(0)> vertical bar vertical bar for all x is an element of E-1 and y is an element of E-2. We also provide some technical examples and counterexamples to illustrate our results.

Název v anglickém jazyce

Hilbert C*-module independence

Popis výsledku anglicky

We introduce the notion of Hilbert C*-module independence: Let A be a unital C*-algebra and let E-i subset of E, i = 1, 2, be ternary subspaces of a Hilbert A-module E. Then, E-1 and E-2 are said to be Hilbert C*-module independent if there are positive constants m and M such that for every state phi(i) on < E-i, E-i >, = 1, 2, there exists a state phi on A such that m phi(i)(vertical bar x vertical bar) <= phi(vertical bar x vertical bar) <= M phi(i)(vertical bar x vertical bar(2))(1/2), for all x is an element of E-i,E- i = 1, 2. We show that it is a natural generalization of the notion of C*-independence of C*-algebras. Moreover, we demonstrate that even in the case of C*-algebras, this concept of independence is new and has a nice characterization in terms of Hahn-Banach-type extensions. We show that if < E-1, E-1 > has the quasi extension property and z is an element of E-1 boolean AND E-2 with vertical bar vertical bar z vertical bar vertical bar = 1, then vertical bar vertical bar z vertical bar vertical bar = 1. Several characterizations of Hilbert C*-module independence and a new characterization of C*-independence are given. One of characterizations states that if z(0) is an element of E-1 boolean AND E-2 is such that < z(0), z(0)> = 1, then E-1 and E-2 are Hilbert C*-module independent if and only if vertical bar vertical bar < x, z(0)> < y, z(0)> vertical bar vertical bar = vertical bar vertical bar < x, z(0)> vertical bar vertical bar vertical bar vertical bar < y, z(0)> vertical bar vertical bar for all x is an element of E-1 and y is an element of E-2. We also provide some technical examples and counterexamples to illustrate our results.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2024

Kód důvěrnosti údajů

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

Název periodika

Mathematische Nachrichten

ISSN

0025-584X

e-ISSN

1522-2616

Svazek periodika

297

Číslo periodika v rámci svazku

2

Stát vydavatele periodika

DE - Spolková republika Německo

Počet stran výsledku

18

Strana od-do

494-511

Kód UT WoS článku

001030304000001

EID výsledku v databázi Scopus

2-s2.0-85164594187