## Journal of the Mathematical Society of Japan

### Completely positive isometries between matrix algebras

Masamichi HAMANA

#### 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.

#### Article information

Source
J. Math. Soc. Japan, Volume 71, Number 2 (2019), 429-449.

Dates
Revised: 17 October 2017
First available in Project Euclid: 25 February 2019

https://projecteuclid.org/euclid.jmsj/1551085234

Digital Object Identifier
doi:10.2969/jmsj/78307830

Mathematical Reviews number (MathSciNet)
MR3943445

Zentralblatt MATH identifier
07090050

#### Citation

HAMANA, Masamichi. Completely positive isometries between matrix algebras. J. Math. Soc. Japan 71 (2019), no. 2, 429--449. doi:10.2969/jmsj/78307830. https://projecteuclid.org/euclid.jmsj/1551085234

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