Maps preserving trace of products of matrices

Abstract

We prove the linearity and injectivity of two maps ${\varphi }_1$ and ${\varphi }_2$ on certain subsets of $M_n$ that satisfy $\mathrm{tr}({\varphi }_1(A){\varphi }_2(B))=\mathrm{tr}(AB)$. We apply it to characterize maps ${\varphi }_i:\mathcal{S}\to \mathcal{S}$ ($i=1,\dots ,m$) satisfying $$\mathrm{tr}({\varphi }_1(A_1)\cdots {\varphi }_m(A_m))=\mathrm{tr}(A_1\cdots A_m)$$ in which $\mathcal{S}$ is the set of $n$-by-$n$ general, Hermitian, or symmetric matrices for $m\ge 3$, or positive definite or diagonal matrices for $m\ge 2$. The real versions are also given.

arXiv comment23 pages, no figure