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, Omega := M-m(D), the m x m matrix algebra over D, and V be a vector space over C. We characterize all n-linear forms on Omega in terms of reduced traces and elementary operators. For m > 1, it is proved that a bilinear form B: Omega x Omega -> V vanishes on zero products of xy and yx if and only if there exist linear maps g, h: Omega -> V such that B(x, y) = g(xy) + h(yx) for all x, y is an element of Omega. As an application, a bilinear form B is completely characterized if B(x, y) = 0 whenever x, y is an element of Omega satisfy xy + xi yx = 0, where xi is a fixed nonzero element in C. (C) 2014 Elsevier Inc. All rights reserved.