LINEAR MAPS PRESERVING KY FAN NORMS AND SCHATTEN NORMS OF TENSOR PRODUCTS OF MATRICES

For a positive integer n, let M-n be the set of n x n complex matrices. Suppose vertical bar vertical bar . vertical bar vertical bar is the Ky Fan k-norm with 1 <= k <= mn or the Schatten p-norm with 1 <= p <= infinity (p not equal 2) on M-mn, where m, n >= 2 are positive integers. It is shown that a linear map phi : M-mn -> M-mn satisfying vertical bar vertical bar A circle times B vertical bar vertical bar = vertical bar vertical bar phi(A circle times B))vertical bar vertical bar for all A is an element of M-m and B is an element of M-n if and only if there are unitary U, V is an element of M-mn such that phi has the form A circle times B bar right arrow U(phi(1)(A) circle times phi(2)(B)) V, where phi(s)(X) is either the identity map X bar right arrow X or the transposition map X bar right arrow X-t. The results are extended to tensor space M-n1 circle times ... circle times M-nm of higher level. The connection of the problem to quantum information science is mentioned.