Hilbert C*-module independence

Result code in 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>

Result on the web

<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>

Result language

angličtina

Original language name

Hilbert C*-module independence

Original language description

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.

Czech name

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Czech description

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Type

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

CEP classification

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OECD FORD branch

10101 - Pure mathematics

Project

—

Continuities

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year

2024

Confidentiality

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

Name of the periodical

Mathematische Nachrichten

ISSN

0025-584X

e-ISSN

1522-2616

Volume of the periodical

297

Issue of the periodical within the volume

2

Country of publishing house

DE - GERMANY

Number of pages

18

Pages from-to

494-511

UT code for WoS article

001030304000001

EID of the result in the Scopus database

2-s2.0-85164594187