Isometries between matrix algebras

As an attempt to understand linear isometries between C*-algebras without the surjectivity assumption, we study linear isometries between matrix algebras. Denote by M-m the algebra of m x m complex matrices. If k greater than or equal to n and phi : M-n --> M-k has the form X --> U[X circle plus f(X)] V or X --> U[X' circle plus f(X)]V for some unitary U, V is an element of M-k and contractive linear map f : M-n --> M-k, then ||phi(X)|| = ||X|| for all X is an element of M-n. We prove that the converse is true if k less than or equal to 2n - 1, and the converse may fail if k greater than or equal to 2n. Related results and questions involving positive linear maps and the numerical range are discussed.