Bilinear forms on matrix algebras vanishing on zero products of xy and yx

Abstract

Let D be a division algebra finite-dimensional over its center C, Ω:=Mm(D), the m×m matrix algebra over D, and V be a vector space over C. We characterize all n-linear forms on Ω in terms of reduced traces and elementary operators. For m>1, it is proved that a bilinear form B:Ω×Ω→V vanishes on zero products of xy and yx if and only if there exist linear maps g,h:Ω→V such that B(x,y)=g(xy)+h(yx) for all x,y∈Ω. As an application, a bilinear form B is completely characterized if B(x,y)=0 whenever x,y∈Ω satisfy xy+ξyx=0, where ξ is a fixed nonzero element in C.