Two-sided zero product properties on symmetric algebras (Q2085425)

Given an associative unital algebra \(A\) over a field \(F\), a bilinear functional \(\phi : A\times A \rightarrow F\) satisfies the two-sided zero product property (2-zpp) if \(xy=yx=0\Rightarrow \phi(xy)=0\). Such a bilinear function is of standard form if \(\phi(x,y)=\tau_1(xy)+\tau_2(yx) \forall x,y\in A\) where \(\tau_1,\tau_2\) are fixed linear functionals. Bilinear functionals of standard form are 2-zpp. A derivation on \(A\) is an additive map \(d : A \rightarrow A\) satisfying \(d(xy)=d(x)y+xd(y)\). A derivation is inner if \(d(x)=[c,x]=cx-xc \forall x\in A\) for some fixed \(c\in A\). The algebra \(A\) is said to be zero product determined (zpd) if every bilinear functional \(\phi : A\times A \rightarrow F\) satisfying \(xy=0\Rightarrow \phi(xy)=0\) is of the form \(\phi(x,y)=\tau(xy)\) where \(\tau\) is a fixed linear functional. The main result of this paper (Theorem 2.1) states that if \(A\) is a zpd associative \(F\)-algebra with nondegenerate trace form \(T\) and \(\phi: A\times A\rightarrow F\) is a 2-zpp bilinear functional, then there exist a unique derivation \(d\) on \(A\) and a unique element \(c\) in \(A\) such that \(\phi(x,y)=T((d(x)+cx)y)\). Furthermore, \(\phi\) is of standard form if and only if \(d\) is inner. The contains several other interesting results and examples.