Completely positive isometries between matrix algebras

Let be a linear map between operator spaces. To measure the intensity of being isometric we associate with it a number, called the isometric degree of and written id (), as follows. Call a strict m-isometry with m a positive integer if it is an m-isometry, but is not an (m+1) -isometry. Define id () to be 0, m, and, respectively if is not an isometry, a strict m-isometry, and a complete isometry, respectively. We show that if: Mₙ Mₚ is a unital completely positive map between matrix algebras, then id () \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 () for some. The ranges of such maps with 1 id () < provide natural examples of operator systems that are isometric, but not completely isometric, to Mₙ. We introduce and classify, up to unital complete isometry, a certain family of such operator systems.