Journal of the Mathematical Society of Japan

Completely positive isometries between matrix algebras

Masamichi HAMANA

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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


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References

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