RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
General information
Latest issue
Forthcoming papers
Archive
Impact factor
Guidelines for authors
License agreement

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Trudy MIAN:
Year:
Volume:
Issue:
Page:
Find






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


Tr. Mat. Inst. Steklova, 2006, Volume 255, Pages 41–54 (Mi tm252)  

This article is cited in 8 scientific papers (total in 8 papers)

Local Convergence in Measure on Semifinite von Neumann Algebras

A. M. Bikchentaev

N. G. Chebotarev Research Institute of Mathematics and Mechanics, Kazan State University

Abstract: Suppose that $\mathcal M$ is a von Neumann algebra of operators on a Hilbert space $\mathcal H$ and $\tau $ is a faithful normal semifinite trace on $\mathcal M$. The set $\widetilde {\mathcal M}$ of all $\tau $-measurable operators with the topology $t_{\tau }$ of convergence in measure is a topological $*$-algebra. The topologies of $\tau $-local and weakly $\tau $-local convergence in measure are obtained by localizing $t_{\tau }$ and are denoted by $t_{\tau \mathrm l}$ and $t_{\mathrm w\tau \mathrm l}$, respectively. The set $\widetilde {\mathcal M}$ with any of these topologies is a topological vector space. The continuity of certain operations and the closedness of certain classes of operators in $\widetilde {\mathcal M}$ with respect to the topologies $t_{\tau \mathrm l}$ and $t_{\mathrm w\tau \mathrm l}$ are proved. S.M. Nikol'skii's theorem (1943) is extended from the algebra $\mathcal B(\mathcal H)$ to semifinite von Neumann algebras. The following theorem is proved: {\itshape For a von Neumann algebra $\mathcal M$ with a faithful normal semifinite trace $\tau $, the following conditions are equivalent\textup : \textup {(i)} the algebra $\mathcal M$ is finite\textup ; \textup {(ii)} $t_{\mathrm w\tau \mathrm l}= t_{\tau \mathrm l}$\textup ; \textup {(iii)} the multiplication is jointly $t_{\tau \mathrm l}$-continuous from $\widetilde {\mathcal M}\times \widetilde {\mathcal M}$ to $\widetilde {\mathcal M}$\textup ; \textup {(iv)} the multiplication is jointly $t_{\mathrm w\tau \mathrm l}$-continuous from $\widetilde {\mathcal M}\times \widetilde {\mathcal M}$ to $\widetilde {\mathcal M}$\textup ; \textup {(v)} the involution is $t_{\tau \mathrm l}$-continuous from $\widetilde {\mathcal M}$ to $\widetilde {\mathcal M}$.}

Full text: PDF file (264 kB)
References: PDF file   HTML file

English version:
Proceedings of the Steklov Institute of Mathematics, 2006, 255, 35–48

Bibliographic databases:

UDC: 517.986+517.987
Received in November 2005

Citation: A. M. Bikchentaev, “Local Convergence in Measure on Semifinite von Neumann Algebras”, Function spaces, approximation theory, and nonlinear analysis, Collected papers, Tr. Mat. Inst. Steklova, 255, Nauka, MAIK Nauka/Inteperiodika, M., 2006, 41–54; Proc. Steklov Inst. Math., 255 (2006), 35–48

Citation in format AMSBIB
\Bibitem{Bik06}
\by A.~M.~Bikchentaev
\paper Local Convergence in Measure on Semifinite von Neumann Algebras
\inbook Function spaces, approximation theory, and nonlinear analysis
\bookinfo Collected papers
\serial Tr. Mat. Inst. Steklova
\yr 2006
\vol 255
\pages 41--54
\publ Nauka, MAIK Nauka/Inteperiodika
\publaddr M.
\mathnet{http://mi.mathnet.ru/tm252}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2301608}
\elib{http://elibrary.ru/item.asp?id=13516363}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2006
\vol 255
\pages 35--48
\crossref{https://doi.org/10.1134/S0081543806040043}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33846881782}


Linking options:
  • http://mi.mathnet.ru/eng/tm252
  • http://mi.mathnet.ru/eng/tm/v255/p41

    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
    Cycle of papers

    This publication is cited in the following articles:
    1. A. M. Bikchentaev, “Local Convergence in Measure on Semifinite von Neumann Algebras, II”, Math. Notes, 82:5 (2007), 703–707  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. Rikchentaev A.M., “Local Convergence in Measure on Semifinite Von Neumann Algebras. III”, Hot Topics in Operator Theory, Conference Proceedings, 2008, 1–12  mathscinet  isi
    3. A. M. Bikchentaev, A. A. Sabirova, “Dominated convergence in measure on semifinite von Neumann algebras and arithmetic averages of measurable operators”, Siberian Math. J., 53:2 (2012), 207–216  mathnet  crossref  mathscinet  isi
    4. A. M. Bikchentaev, “Concerning the Theory of $\tau$-Measurable Operators Affiliated to a Semifinite von Neumann Algebra”, Math. Notes, 98:3 (2015), 382–391  mathnet  crossref  crossref  mathscinet  isi  elib
    5. A. M. Bikchentaev, “Convergence of integrable operators affiliated to a finite von Neumann algebra”, Proc. Steklov Inst. Math., 293 (2016), 67–76  mathnet  crossref  crossref  mathscinet  isi  elib
    6. A. M. Bikchentaev, “On Idempotent $\tau$-Measurable Operators Affiliated to a von Neumann Algebra”, Math. Notes, 100:4 (2016), 515–525  mathnet  crossref  crossref  mathscinet  isi  elib
    7. M. A. Muratov, V. I. Chilin, “Topologicheskie algebry izmerimykh i lokalno izmerimykh operatorov”, Trudy Krymskoi osennei matematicheskoi shkoly-simpoziuma, SMFN, 61, RUDN, M., 2016, 115–163  mathnet
    8. Bikchentaev A.M., “Trace and integrable operators affiliated with a semifinite von Neumann algebra”, Dokl. Math., 93:1 (2016), 16–19  crossref  mathscinet  zmath  isi  elib  scopus
  •    . . .  Proceedings of the Steklov Institute of Mathematics
    Number of views:
    This page:598
    Full text:327
    References:205

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2020