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

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 $\widetilde {\mathcal M}$.}

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

Bibliographic databases:

UDC: 517.986+517.987

Citation: A. M. Bikchentaev, “Local Convergence in Measure on Semifinite von Neumann Algebras”, Function spaces, approximation theory, and nonlinear analysis, Collected papers, Trudy 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 Trudy 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{https://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{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33846881782} 

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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
2. Rikchentaev A.M., “Local Convergence in Measure on Semifinite Von Neumann Algebras. III”, Hot Topics in Operator Theory, Conference Proceedings, 2008, 1–12
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
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
5. A. M. Bikchentaev, “Convergence of integrable operators affiliated to a finite von Neumann algebra”, Proc. Steklov Inst. Math., 293 (2016), 67–76
6. A. M. Bikchentaev, “On Idempotent $\tau$-Measurable Operators Affiliated to a von Neumann Algebra”, Math. Notes, 100:4 (2016), 515–525
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
8. Bikchentaev A.M., “Trace and integrable operators affiliated with a semifinite von Neumann algebra”, Dokl. Math., 93:1 (2016), 16–19
9. Bikchentaev A.M., “Rearrangements of Tripotents and Differences of Isometries in Semifinite Von Neumann Algebras”, Lobachevskii J. Math., 40:10, SI (2019), 1450–1454
10. A. M. Bikchentaev, “Convergence in measure and $\tau$-compactness of $\tau$-measurable operators, affiliated with a semifinite von Neumann algebra”, Russian Math. (Iz. VUZ), 64:5 (2020), 79–82
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