Isometries of the Toeplitz Matrix Algebra

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We study the structure of isometries defined on the algebra A of upper-triangular Toeplitz matrices. Our first result is that a continuous multiplicative isometry A→ Mₙ must be of the form either A UAU* or A U AU*, 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.