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ZERO JORDAN PRODUCT DETERMINED BANACH ALGEBRAS
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 111 / Issue 2 / October 2021
- Published online by Cambridge University Press:
- 08 January 2020, pp. 145-158
- Print publication:
- October 2021
-
- Article
- Export citation
-
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.
