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A Banach algebra $A$ is said to be zero Lie product determined if every continuous bilinear functional $varphi colon Atimes Ato mathbb{C}$ satisfying $varphi(a,b)=0$ whenever $ab=ba$ is of the form $varphi(a,b)=omega(ab-ba)$ for some $omegain A^*$. We prove that $A$ has this property provided that any of the following three conditions holds: (i) $A$ is a weakly amenable Banach algebra with property $mathbb{B}$ and having a bounded approximate identity, (ii) every continuous cyclic Jordan derivation from $A$ into $A^*$ is an inner derivation, (iii) $A$ is the algebra of all $ntimes n$ matrices, where $nge 2$, over a cyclically amenable Banach algebra with a bounded approximate identity.
A Banach algebra $A$ is said to be zero Lie product determined if every continuous bilinear functional $varphi colon Atimes Ato mathbb{C}$ with the property that $varphi(a,b)=0$ whenever $a$ and $b$ commute is of the form $varphi(a,b)=tau(ab-ba)$ for
A Banach algebra $A$ is said to be a zero Jordan product determined Banach algebra if every continuous bilinear map $varphicolon Atimes Ato X$, where $X$ is an arbitrary Banach space, which satisfies $varphi(a,b)=0$ whenever $a$, $bin A$ are such tha
$C^*$-algebras, group algebras, and the algebra $mathcal{A}(X)$ of approximable operators on a Banach space $X$ having the bounded approximation property are known to be zero product determined. We are interested in giving a quantitative estimate o
A Lie algebra $L$ over a field $mathbb{F}$ is said to be zero product determined (zpd) if every bilinear map $f:Ltimes Lto mathbb{F}$ with the property that $f(x,y)=0$ whenever $x$ and $y$ commute is a coboundary. The main goal of the paper is to det
We introduce the class of slicely countably determined Banach spaces which contains in particular all spaces with the RNP and all spaces without copies of $ell_1$. We present many examples and several properties of this class. We give some applicatio