## Journal of the Mathematical Society of Japan

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

### 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.

#### Article information

**Source**

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

**Dates**

Received: 21 June 2017

Revised: 17 October 2017

First available in Project Euclid: 25 February 2019

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.2969/jmsj/78307830

**Mathematical Reviews number (MathSciNet)**

MR3943445

**Zentralblatt MATH identifier**

07090050

**Subjects**

Primary: 46L07: Operator spaces and completely bounded maps [See also 47L25]

Secondary: 46B04: Isometric theory of Banach spaces

**Keywords**

completely positive isometry matrix algebra operator system

#### 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