ZERO JORDAN PRODUCT DETERMINED BANACH ALGEBRAS

A Banach algebra $A$ is said to be a zero Jordan product determined Banach algebra if, for every Banach space $X$ , every bilinear map $\unicode[STIX]{x1D711}:A\times A\rightarrow X$ satisfying $\unicode[STIX]{x1D711}(a,b)=0$ whenever $a$ , $b\in A$ are such that $ab+ba=0$ , is of the form $\unicode...

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Bibliographic Details
Published in:Journal of the Australian Mathematical Society (2001) 2021-10, Vol.111 (2), p.145-158
Main Authors: ALAMINOS, J., BREŠAR, M., EXTREMERA, J., VILLENA, A. R.
Format: Article
Language:English
Subjects:
Algebra
Banach spaces
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Summary:A Banach algebra $A$ is said to be a zero Jordan product determined Banach algebra if, for every Banach space $X$ , every bilinear map $\unicode[STIX]{x1D711}:A\times A\rightarrow X$ satisfying $\unicode[STIX]{x1D711}(a,b)=0$ whenever $a$ , $b\in A$ are such that $ab+ba=0$ , is of the form $\unicode[STIX]{x1D711}(a,b)=\unicode[STIX]{x1D70E}(ab+ba)$ for some continuous linear map $\unicode[STIX]{x1D70E}$ . We show that all $C^{\ast }$ -algebras and all group algebras $L^{1}(G)$ of amenable locally compact groups have this property and also discuss some applications.
ISSN:1446-7887
1446-8107
DOI:10.1017/S1446788719000478