Let Φ be a linear map between operator spaces. To measure the intensity of Φ being isometric we associate with it a number, called the isometric degree of Φ and written id(Φ), as follows. Call Φ a strict m-isometry with m a positive integer if it is an m-isometry, but is not an (m+ 1)-isometry. Define id(Φ) to be 0, m, and ∞, respectively if Φ is not an isometry, a strict m-isometry, and a complete isometry, respectively. We show that if Φ : M_n → M_p is a unital completely positive map between matrix algebras, then id(Φ) ∈ {0, 1, 2, .. ., [(n - 1)/2], ∞} and that when n ≥ 3 is fixed and p is sufficiently large, the values 1, 2, . . ., [(n - 1)/2] are attained as id(Φ) for some Φ. The ranges of such maps Φ with 1 ≤ id(Φ) < ∞ provide natural examples of operator systems that are isometric, but not completely isometric, to M_n. We introduce and classify, up to unital complete isometry, a certain family of such operator systems.
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