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 Vladikavkaz. Mat. Zh., 2019, Volume 21, Number 4, Pages 5–10 (Mi vmj702)

$2$-Local isometries of non-commutative Lorentz spaces

A. A. Alimova, V. I. Chilinb

a Tashkent Institute of Design, Construction and Maintenance of Automobile Roads, 20 Amir Temur Av., Tashkent 100060, Uzbekistan
b National University of Uzbekistan, Vuzgorodok, Tashkent 100174, Uzbekistan

Abstract: Let $\mathcal M$ be a von Neumann algebra equipped with a faithful normal finite trace $\tau$, and let $S( \mathcal{M}, \tau)$ be an $\ast$-algebra of all $\tau$-measurable operators affiliated with $\mathcal M$. For $x \in S( \mathcal{M}, \tau)$ the generalized singular value function $\mu(x):t\rightarrow \mu(t;x)$, $t>0$, is defined by the equality $\mu(t;x)=\inf\{\|xp\|_{\mathcal{M}}: p^2=p^*=p \in \mathcal{M}, \tau(\mathbf{1}-p)\leq t\}.$ Let $\psi$ be an increasing concave continuous function on $[0, \infty)$ with $\psi(0) = 0$, $\psi(\infty)=\infty$, and let $\Lambda_\psi(\mathcal M,\tau) = \{x \in S( \mathcal{M}, \tau): \| x \|_{\psi} =\int_0^{\infty}\mu(t;x)d\psi(t) < \infty \}$ be the non-commutative Lorentz space. A surjective (not necessarily linear) mapping $V: \Lambda_\psi(\mathcal M,\tau) \to \Lambda_\psi(\mathcal M,\tau)$ is called a surjective $2$-local isometry, if for any $x, y \in \Lambda_\psi(\mathcal M,\tau)$ there exists a surjective linear isometry $V_{x, y}: \Lambda_\psi(\mathcal M,\tau) \to \Lambda_\psi(\mathcal M,\tau)$ such that $V(x) = V_{x, y}(x)$ and $V(y) = V_{x, y}(y)$. It is proved that in the case when $\mathcal{M}$ is a factor, every surjective $2$-local isometry $V:\Lambda_\psi(\mathcal M,\tau) \to \Lambda_\psi(\mathcal M,\tau)$ is a linear isometry.

Key words: measurable operator, Lorentz space, isometry.

DOI: https://doi.org/10.23671/VNC.2019.21.44595

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UDC: 517.98
MSC: 46L52, 46B04
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Citation: A. A. Alimov, V. I. Chilin, “$2$-Local isometries of non-commutative Lorentz spaces”, Vladikavkaz. Mat. Zh., 21:4 (2019), 5–10

Citation in format AMSBIB
\Bibitem{AliChi19} \by A.~A.~Alimov, V.~I.~Chilin \paper $2$-Local isometries of non-commutative Lorentz spaces \jour Vladikavkaz. Mat. Zh. \yr 2019 \vol 21 \issue 4 \pages 5--10 \mathnet{http://mi.mathnet.ru/vmj702} \crossref{https://doi.org/10.23671/VNC.2019.21.44595} 

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