Isometries of the Toeplitz matrix algebra

We study the structure of isometries defined on the algebra A of upper-triangular Toeplitz matrices. Our first result is that a continuous multiplicative norm preserving map A→Mn(C) must be of the form either A↦UAU⁎ or A↦UA‾U⁎, where A‾ is the complex conjugation and U is a unitary matrix. In our second result we use a range of ideas in operator theory and linear algebra to show that every linear isometry A→Mn(C) is of the form A↦UAV where U and V are two unitary matrices. This implies, in particular, that every such an isometry is a complete isometry and that a unital linear isometry A→Mn(C) is necessarily an algebra homomorphism.