Isometries Between Matrix Algebras
نویسندگان
چکیده
As an attempt to understand linear isometries between C∗-algebras without the surjectivity assumption, we study linear isometries between matrix algebras. Denote by Mm the algebra of m×m complex matrices. If k ≥ n and φ : Mn → Mk has the form X 7→ U [X ⊕ f(X)]V or X 7→ U [X t ⊕ f(X)]V for some unitary U, V ∈ Mk and contractive linear map f : Mn → Mk, then ‖φ(X)‖ = ‖X‖ for all X ∈ Mn. We prove that the converse is true if k ≤ 2n− 1, and the converse may fail if k ≥ 2n. Related results and questions involving positive linear maps and the numerical range are discussed. 2000 Mathematics Subject Classifications: 15A04, 15A60.
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