## Rocky Mountain Journal of Mathematics

### Maps preserving quasi-isometries on Hilbert $C^*$-modules

#### Abstract

Let $\mathcal {K}(\mathcal {H})$ be the $C^*$-algebra of compact op\-erators on a Hilbert space $\mathcal {H}$. Let $E$ be a Hilbert $\mathcal {K}(\mathcal {H})$-mod\-ule and $\mathcal {L}(E)$ the $C^*$-algebra of all adjointable maps on $E$. In this paper, we prove that, if $\varphi :\mathcal {L}(E)\to \mathcal {L}(E)$ is a unital surjective bounded linear map, which preserves quasi-isometries in both directions, then there are unitary oper\-ators $U, V\in \mathcal {L}(E)$ such that $\varphi (T)=UTV\quad \mbox {or}\quad \varphi (T)=UT^{tr }V$ for all $T\in \mathcal {L}(E)$, where $T^{tr }$ is the transpose of $T$ with re\-spect to an arbitrary but fixed orthonormal basis of $E$.

#### Article information

Source
Rocky Mountain J. Math., Volume 48, Number 4 (2018), 1219-1229.

Dates
First available in Project Euclid: 30 September 2018

https://projecteuclid.org/euclid.rmjm/1538272831

Digital Object Identifier
doi:10.1216/RMJ-2018-48-4-1219

Mathematical Reviews number (MathSciNet)
MR3859756

Zentralblatt MATH identifier
06958777

Subjects
Primary: 46L05: General theory of $C^*$-algebras 46L08: $C^*$-modules

#### Citation

Majidi, Alireza; Amyari, Maryam. Maps preserving quasi-isometries on Hilbert $C^*$-modules. Rocky Mountain J. Math. 48 (2018), no. 4, 1219--1229. doi:10.1216/RMJ-2018-48-4-1219. https://projecteuclid.org/euclid.rmjm/1538272831

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