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.
@article{bwmeta1.element.doi-10_2478_s11533-011-0136-6,
author = {Shavkat Ayupov and Karimbergen Kudaybergenov and Rauaj Djumamuratov},
title = {Topologies on central extensions of von Neumann algebras},
journal = {Open Mathematics},
volume = {10},
year = {2012},
pages = {656-664},
zbl = {1250.46039},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0136-6}
}
Shavkat Ayupov; Karimbergen Kudaybergenov; Rauaj Djumamuratov. Topologies on central extensions of von Neumann algebras. Open Mathematics, Tome 10 (2012) pp. 656-664. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0136-6/
[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] 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 | Zbl 1175.46054
[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 | Zbl 1246.46058
[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 | Zbl 1203.46047
[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 | Zbl 1263.46057
[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) | Zbl 1199.47002
[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) | Zbl 1199.46136
[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] Sarymsakov T.A., Ayupov Sh.A, Khadzhiev Dzh., Chilin V.I., Ordered Algebras, FAN, Tashkent, 1983 (in Russian)
[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 | Zbl 0051.34201
[11] Yeadon F.J., Convergence of measurable operators, Proc. Cambridge Philos. Soc., 1973, 74, 257–268 http://dx.doi.org/10.1017/S0305004100048052 | Zbl 0272.46043