Completely positive isometries between matrix algebras

Abstract

Let $\varphi$ be a linear map between operator spaces. To measure the intensity of $\varphi$ being isometric we associate with it a number, called the isometric degree of $\varphi$ and written $\mathrm{id}(\varphi)$, as follows. Call $\varphi$ a strict $m$-isometry with $m$ a positive integer if it is an $m$-isometry, but is not an $(m+1)$-isometry. Define $\mathrm{id}(\varphi)$ to be 0, $m$, and $\infty$, respectively if $\varphi$ is not an isometry, a strict $m$-isometry, and a complete isometry, respectively. We show that if $\varphi:M_n\to M_p$ is a unital completely positive map between matrix algebras, then $\mathrm{id}(\varphi) \in \{0,\,1,\,2,\,\dots,\,[({n-1})/{2}],\,\infty\}$ and that when $n\ge 3$ is fixed and $p$ is sufficiently large, the values $1,\,2,\,\dots,\,[({n-1})/{2}]$ are attained as $\mathrm{id}(\varphi)$ for some $\varphi$. The ranges of such maps $\varphi$ with $1 \le \mathrm{id}(\varphi)<\infty$ 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.