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ZERO JORDAN PRODUCT DETERMINED BANACH ALGEBRAS

Part of: Abstract harmonic analysis Selfadjoint operator algebras Topological algebras, normed rings and algebras, Banach algebras

Published online by Cambridge University Press:  08 January 2020

J. ALAMINOS ,
M. BREŠAR ,
J. EXTREMERA  and
A. R. VILLENA
J. ALAMINOS
Affiliation:
Departamento de Análisis, Matemático, Facultad de Ciencias, Universidad de Granada, 18071Granada, Spain e-mail: alaminos@ugr.es
M. BREŠAR
Affiliation:
Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000Ljubljana, Slovenia Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška 160, 2000Maribor, Slovenia e-mail: matej.bresar@fmf.uni-lj.si
J. EXTREMERA
Affiliation:
Departamento de Análisis, Matemático, Facultad de Ciencias, Universidad de Granada, 18071Granada, Spain e-mail: jlizana@ugr.es
A. R. VILLENA
Affiliation:
Departamento de Análisis, Matemático, Facultad de Ciencias, Universidad de Granada, 18071Granada, Spain e-mail: avillena@ugr.es
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Abstract

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.

Keywords

C∗ -algebragroup algebrazero Jordan product determined Banach algebrazero product determined Banach algebrasymmetrically amenable Banach algebraweakly amenable Banach algebra

MSC classification

Primary: 43A20: $L^1$-algebras on groups, semigroups, etc.
Secondary: 46H05: General theory of topological algebras 46L05: General theory of $C^*$-algebras
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 111 , Issue 2 , October 2021 , pp. 145 - 158
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by A. Sims

The authors were supported by MINECO grant PGC2018-093794-B-I00. The first, the third and the fourth named authors were supported by Junta de Andalucía grant FQM-185. The second named author was supported by ARRS grant P1-0288.

References

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