Completely positive isometries between matrix algebras

Abstract

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.