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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 determine whether or not some important Lie algebras are zpd. We show that the Galilei Lie algebra $mathfrak{sl}_2ltimes V$, where $V$ is a simple $mathfrak{sl}_2$-module, is zpd if and only if $dim V =2$ or $dim V$ is odd. The class of zpd Lie algebras also includes the quantum torus Lie algebras $mathcal{L}_q$ and $mathcal{L}^+_q$, the untwisted affine Lie algebras, the Heisenberg Lie algebras, and all Lie algebras of dimension at most $3$, while the class of non-zpd Lie algebras includes the ($4$-dimensional) aging Lie algebra $mathfrak {age}(1)$ and all Lie algebras of dimension more than $3$ in which only linearly dependent elements commute. We also give some evidence of the usefulness of the concept of a zpd Lie algebra by using it in the study of commutativity preserving linear maps.
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A Banach algebra $A$ is said to be zero Lie product determined if every continuous bilinear functional $varphi colon Atimes Ato mathbb{C}$ with the property that $varphi(a,b)=0$ whenever $a$ and $b$ commute is of the form $varphi(a,b)=tau(ab-ba)$ for some $tauin A^*$. In the first part of the paper we give some general remarks on this class of algebras. In the second part we consider amenable Banach algebras and show that all group algebras $L^1(G)$ with $G$ an amenable locally compact group are zero Lie product determined.
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.
We prove that free pre-Lie algebras, when considered as Lie algebras, are free. Working in the category of S-modules, we define a natural filtration on the space of generators. We also relate the symmetric group action on generators with the structure of the anticyclic PreLie operad.
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 that $ab+ba=0$, is of the form $varphi(a,b)=sigma(ab+ba)$ for some continuous linear map $sigma$. We show that all $C^*$-algebras and all group algebras $L^1(G)$ of amenable locally compact groups have this property, and also discuss some applications.
$C^*$-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 of this property by finding, for each Banach algebra $A$ of the above classes, a constant $alpha$ with the property that for every continuous bilinear functional $varphicolon A times Atomathbb{C}$ there exists a continuous linear functional $xi$ on $A$ such that [ sup_{Vert aVert=Vert bVert=1}vertvarphi(a,b)-xi(ab)vertle alphasup_{mathclap{substack{Vert aVert=Vert bVert=1, ab=0}}}vertvarphi(a,b)vert. ]
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