Linear rank one preservers between spaces of matrices with zero trace (Q1020929)

Suppose \(X\) and \(Y\) are at least two-dimensional vector spaces over a field \(F\), and equipped with a bilinear form \(\langle.,.\rangle: X\times Y\to F\) such that \(\langle x,Y\rangle=0\) implies \(x=0\). This naturally induces a trace-like linear functional Tr\(:X\otimes Y\to F\) with Tr\((x\otimes y)=\langle x,y\rangle\). The main result then classifies linear maps \(T\) from the kernel \((X\otimes Y)_0\) of Tr into another tensor product space \(U\otimes V\) which preserve nonzero decomposable tensors. When \(\dim Y\geq 4\) it is shown that either the image of \(T\) is a subspace of decomposable tensors or else \(T=f\otimes g\), possibly composed with a map that swaps tensors on \(X\otimes Y\) i.e., \(x\otimes y\mapsto y\otimes x\), for some pair of injective linear maps \(f\) and \(g\). A similar result is verified also when \(2\leq\dim Y\leq 3\) and the field \(F\) contains at least four elements. The diverse applications vary from classifying linear rank-one preservers between rectangular, i.e., \(M_{m\times n}(F)\) for \(m\leq n\), matrices with zero sum of diagonal entries on one hand, to classifying (for any \(n\) and in any field with only two minor exceptions) bilinear functions \(\theta: M_n(F)\times M_n(F)\to M_n(F)\) subject to \[ AB+\lambda BA=0\Longrightarrow\theta(A,B)=0 \] and \[ \text{rank}\,(AB+\lambda BA)=1\Longrightarrow\text{rank}\,\theta(A,B)=1, \] on the other. The paper is accompanied by multitude of examples which show the optimality of the obtained results.