Two-sided zero product properties on symmetric algebras

Abstract

We characterize bilinear functionals ϕ on a symmetric algebra A satisfying the two-sided zero product property (the 2-zpp, i.e., ϕ(x,y)=0 whenever xy=yx=0). If A is also a zero product determined algebra and if every derivation of the algebra A is inner, then A is a 2-zpd algebra (i.e., every bilinear functional on A satisfying the 2-zpp is of the form (x,y)↦τ1(xy)+τ2(yx) for x,y∈A, where τ1,τ2 are linear functionals on A). Conversely, if A is a finite-dimensional 2-zpd algebra, then the derivations of A are characterized, that is, given any derivation d of the algebra A, there exists a∈A such that, for all x∈A, d(x)−[a,x] lies in the Jacobson radical of A. Finally, we determine all bilinear functionals satisfying the 2-zpp on a specific zpd symmetric algebra and hence decide whether such an algebra is 2-zpd.