Maps preserving trace of products of matrices

We prove the linearity and injectivity of two maps phi(1) and phi(2) on certain subsets of M-n (C) that satisfy tr(phi(1)(A)phi(2)(B)) = tr(AB) for all A and B in the domain. We apply it to characterize maps phi(i) : S -> S (i = 1, ..., m) satisfying tr(phi(1) (A(1)) . . . phi(m)(A(m))) = tr(A(1). . .A(m)) in which S is the set of n-by-n general, Hermitian, or symmetric matrices for m >= 3, or positive definite or diagonal matrices for m >= 2. The real versions are presented. The results also characterize maps on these sets S that preserve the multiplicative spectrum.