Topologies on central extensions of von Neumann algebras

Shavkat Ayupov; Karimbergen Kudaybergenov; Rauaj Djumamuratov

Open Mathematics (2012)

  • Volume: 10, Issue: 2, page 656-664
  • ISSN: 2391-5455

Abstract

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Given a von Neumann algebra M, we consider the central extension E(M) of M. We introduce the topology t c(M) on E(M) generated by a center-valued norm and prove that it coincides with the topology of local convergence in measure on E(M) if and only if M does not have direct summands of type II. We also show that t c(M) restricted to the set E(M)h of self-adjoint elements of E(M) coincides with the order topology on E(M)h if and only if M is a σ-finite type Ifin von Neumann algebra.

How to cite

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Shavkat Ayupov, Karimbergen Kudaybergenov, and Rauaj Djumamuratov. "Topologies on central extensions of von Neumann algebras." Open Mathematics 10.2 (2012): 656-664. <http://eudml.org/doc/269809>.

@article{ShavkatAyupov2012,
abstract = {Given a von Neumann algebra M, we consider the central extension E(M) of M. We introduce the topology t c(M) on E(M) generated by a center-valued norm and prove that it coincides with the topology of local convergence in measure on E(M) if and only if M does not have direct summands of type II. We also show that t c(M) restricted to the set E(M)h of self-adjoint elements of E(M) coincides with the order topology on E(M)h if and only if M is a σ-finite type Ifin von Neumann algebra.},
author = {Shavkat Ayupov, Karimbergen Kudaybergenov, Rauaj Djumamuratov},
journal = {Open Mathematics},
keywords = {von Neumann algebras; Central extensions; Local measure topology; central extensions; local measure topology},
language = {eng},
number = {2},
pages = {656-664},
title = {Topologies on central extensions of von Neumann algebras},
url = {http://eudml.org/doc/269809},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Shavkat Ayupov
AU - Karimbergen Kudaybergenov
AU - Rauaj Djumamuratov
TI - Topologies on central extensions of von Neumann algebras
JO - Open Mathematics
PY - 2012
VL - 10
IS - 2
SP - 656
EP - 664
AB - Given a von Neumann algebra M, we consider the central extension E(M) of M. We introduce the topology t c(M) on E(M) generated by a center-valued norm and prove that it coincides with the topology of local convergence in measure on E(M) if and only if M does not have direct summands of type II. We also show that t c(M) restricted to the set E(M)h of self-adjoint elements of E(M) coincides with the order topology on E(M)h if and only if M is a σ-finite type Ifin von Neumann algebra.
LA - eng
KW - von Neumann algebras; Central extensions; Local measure topology; central extensions; local measure topology
UR - http://eudml.org/doc/269809
ER -

References

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  1. [1] Albeverio S., Ayupov Sh.A., Kudaybergenov K.K., Derivations on the algebra of measurable operators affiliated with a type I von Neumann algebra, Siberian Adv. Math., 2008, 18(2), 86–94 http://dx.doi.org/10.3103/S1055134408020028 
  2. [2] Albeverio S., Ayupov Sh.A., Kudaybergenov K.K., Structure of derivations on various algebras of measurable operators for type I von Neumann algebras, J. Func. Anal., 2009, 256(9), 2917–2943 http://dx.doi.org/10.1016/j.jfa.2008.11.003 Zbl1175.46054
  3. [3] Albeverio S., Ayupov Sh.A., Kudaybergenov K.K., Djumamuratov R.T., Automorphisms of central extensions of type I von Neumann algebras, Studia Math. (in press), preprint available at http://arxiv.org/abs/1104.4698 Zbl1246.46058
  4. [4] Ayupov Sh.A., Kudaybergenov K.K., Derivations on algebras of measurable operators, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 2010, 13(2), 305–337 http://dx.doi.org/10.1142/S0219025710004085 Zbl1203.46047
  5. [5] Ayupov Sh.A., Kudaybergenov K.K., Additive derivations on algebras of measurable operators, J. Operator Theory (in press), preprint available at http://users.ictp.it/_pub_off/preprints-sources/2009/IC2009059P.pdf Zbl1263.46057
  6. [6] Muratov M.A., Chilin V.I., Algebras of Measurable and Locally Measurable Operators, Proceedings of Institute of Mathematics, Ukrainian Academy of Sciences, 69, Kiev, 2007 (in Russian) Zbl1199.47002
  7. [7] Muratov M.A., Chilin V.I., Central extensions of *-algebras of measurable operators, Reports of the National Academy of Sciences of Ukraine, 2009, 7, 24–28 (in Russian) Zbl1199.46136
  8. [8] Muratov M.A., Chilin V.I., (o)-Topology in *-algebras of locally measurable operators, Ukrainian Math. J., 2009, 61(11), 1798–1808 http://dx.doi.org/10.1007/s11253-010-0313-y 
  9. [9] Sarymsakov T.A., Ayupov Sh.A, Khadzhiev Dzh., Chilin V.I., Ordered Algebras, FAN, Tashkent, 1983 (in Russian) 
  10. [10] Segal I.E., A non-commutative extension of abstract integration, Ann. Math., 1953, 57(3), 401–457 http://dx.doi.org/10.2307/1969729 Zbl0051.34201
  11. [11] Yeadon F.J., Convergence of measurable operators, Proc. Cambridge Philos. Soc., 1973, 74, 257–268 http://dx.doi.org/10.1017/S0305004100048052 Zbl0272.46043

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