Vladikavkazskii Matematicheskii Zhurnal
RUS  ENG    JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB  
General information
Latest issue
Archive
Impact factor

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Vladikavkaz. Mat. Zh.:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


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

Full text: PDF file (202 kB)
References: PDF file   HTML file

UDC: 517.98
MSC: 46L52, 46B04
Received: 20.06.2019
Language:

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}


Linking options:
  • http://mi.mathnet.ru/eng/vmj702
  • http://mi.mathnet.ru/eng/vmj/v21/i4/p5

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles
  • Владикавказский математический журнал
    Number of views:
    This page:77
    Full text:22
    References:5

     
    Contact us:
     Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2022