Geodesic
Minimality of Convergence in Measure Topologies on Finite von Neumann Algebras
Matematičeskie zametki, Tome 75 (2004) no. 3, pp. 342-349.

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We prove that the natural embedding of the metric ideal space on a finite von Neumann algebra ${\mathscr M}$ into the $*$-algebra of measurable operators $\widetilde {\mathscr M}$ endowed with the topology of convergence in measure is continuous. Using this fact, we prove that the topology of convergence in measure is a minimal one among all metrizable topologies consistent with the ring structure on $\widetilde {\mathscr M}$. 
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A. M. Bikchentaev. Minimality of Convergence in Measure Topologies on Finite von Neumann Algebras. Matematičeskie zametki, Tome 75 (2004) no. 3, pp. 342-349. http://geodesic.mathdoc.fr/item/MZM_2004_75_3_a1/

[1] Trunov N. V., Sherstnev A. N., “Vvedenie v teoriyu nekommutativnogo integrirovaniya”, Itogi nauki i tekhniki. Sovremennye problemy matematiki. Noveishie dostizheniya, 27, VINITI, M., 1985, 167–190 | MR

[2] Kosaki H., “On the continuiti of the map $\varphi \mapsto |\varphi|$ from the predual of $W^{\ast}$-algebra”, J. Funct. Anal., 59:1 (1984), 123–131 | DOI | MR | Zbl

[3] Kakutani S., “Über die Metrization der topologischen Gruppen”, Proc. Imp. Acad. Japan, 12:4 (1936), 82–84 | DOI | MR | Zbl

[4] Nelson E., “Notes on non-commutative integration”, J. Funct. Anal., 15:2 (1974), 103–116 | DOI | MR | Zbl

[5] Bikchentaev A. M., $F$-normirovannye idealnye prostranstva izmerimykh operatorov, Rukopis dep. v VINITI 4 iyulya 1988 g., No 5375-V88 Dep., Kazan. un-t, Kazan, 1988

[6] Yeadon F. J., “Non-commutative $L^p$-spaces”, Math. Proc. Cambridge Phil. Soc., 77:1 (1975), 91–102 | DOI | MR | Zbl

[7] Bikchentaev A. M., “On noncommutative function spaces”, Amer. Math. Soc. Transl. (2), 154 (1992), 179–187

[8] Ovchinnikov V. I., “Simmetrichnye prostranstva izmerimykh operatorov”, Dokl. AN SSSR, 191:4 (1970), 769–771 | MR | Zbl

[9] Ciach L. J., “Linear-topological spaces of operators affiliated with a von Neumann algebra”, Bull. Acad. Pol. Sci. Math., 31:3–4 (1983), 161–166 | MR | Zbl

[10] Tikhonov O. E., “On the Losanovskiĭ class of condensing operators and its applications to non-commutative integration”, Israel Mathematical Conference Proceedings, 13 (1999), 204–208 | MR | Zbl

[11] Takesaki M., Theory of operator algebras, V. I, Springer-Verlag, New York–Heidelberg–Berlin, 1979 | MR

[12] Szeptycki P., “On solid spaces of measurable functions”, Bull. Acad. Pol. Sci. Math., 30:3–4 (1982), 115–116 | MR | Zbl

[13] Kaplansky I., “Projections in Banach algebras”, Ann. Math., 53:2 (1951), 235–249 | DOI | MR | Zbl

[14] Peck N. T., “On non locally convex spaces, II”, Math. Annalen, 178:3 (1968), 209–218 | DOI | MR | Zbl

[15] Yeadon F. J., “Convergence of measurable operators”, Proc. Cambridge Phil. Soc., 74:2 (1973), 257–268 | DOI | MR | Zbl

[16] Wnuk W., “On a continuous embedding into a space of measurable functions”, Bull. Acad. Pol. Sci. Math., 34:7–8 (1986), 413–416 | MR | Zbl

[17] Muratov M. A., “Skhodimosti v koltse izmerimykh operatorov”, Funktsionalnyi analiz, Sbor. nauch. tr. Tashkent. un-ta, no. 573, Izd-vo TashGU, Tashkent, 1978, 51–58 | MR