Local Convergence in Measure on Semifinite von Neumann Algebras

A. M. Bikchentaev

Informatics and Automation, Function spaces, approximation theory, and nonlinear analysis, Tome 255 (2006), pp. 41-54.

Résumé

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 $\widetilde {\mathcal M}$.}

Bibliographie

[1] Bikchentaev A.M., “O minimalnosti topologii skhodimosti po mere na konechnykh algebrakh fon Neimana”, Mat. zametki, 75:3 (2004), 342–349 | MR | Zbl

[2] Bikchentaev A.M., “The continuity of multiplication for two topologies associated with a semifinite trace on von Neumann algebra”, Lobachevskii J. Math., 14 (2004), 17–24 | MR | Zbl

[3] Kadison R.V., Ringrose J.R., Fundamentals of the theory of operator algebras, vol. 1, Acad. Press, New York; London; Paris, 1983, 398 pp. | MR | Zbl

[4] Gribanov Yu.I., “O metrizatsii odnogo prostranstva funktsii”, Comment. Math. Univ. Carolinae., 4:1 (1963), 43–46 | MR | Zbl

[5] Nikolskii S.M., “Lineinye uravneniya v lineinykh normirovannykh prostranstvakh”, Izv. AN SSSR. Ser. mat., 7:3 (1943), 147–166

[6] Takesaki M., Theory of operator algebras, vol. 1, Springer, New York; Heidelberg; Berlin, 1979, 415 pp. | MR

[7] Segal I.E., “A non-commutative extension of abstract integration”, Ann. Math., 57:3 (1953), 401–457 | DOI | MR | Zbl

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

[9] Terp M., $L^p$-spaces associated with von Neumann algebras, Copenhagen Univ., Copenhagen, 1981, 100 pp.

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

[11] Fack T., Kosaki H., “Generalized $s$-numbers of $\tau $-measurable operators”, Pacif. J. Math., 123:2 (1986), 269–300 | MR | Zbl

[12] Ovchinnikov V.I., “O $s$-chislakh izmerimykh operatorov”, Funkts. anal. i ego pril., 4:3 (1970), 78–85 | MR | Zbl

[13] Bikchentaev A.M., “On noncommutative function spaces”, Selected papers in $K$-theory, AMS Transl. Ser. 2, 154, Amer. Math. Soc., Providence (RI), 1992, 179–187

[14] Danford N., Shvarts Dzh.T., Lineinye operatory: Obschaya teoriya, Izd-vo inostr. lit., M., 1962, 874 pp.

[15] Skvortsova G.Sh., Tikhonov O.E., “Vypuklye mnozhestva v nekommutativnykh $L^1$-prostranstvakh, zamknutye v topologii lokalnoi skhodimosti po mere”, Izv. vuzov. Matematika, 1998, no. 8, 48–55 | MR | Zbl

[16] Dodds P.G., Dodds T.K., Sukochev F.A., Tikhonov O.Ye., “A non-commutative Yosida–Hewitt theorem and convex sets of measurable operators closed locally in measure”, Positivity, 9:3 (2005), 457–484 | DOI | MR | Zbl

[17] Ciach L.J., “Some remarks on the convergence in measure and on a dominated sequence of operators measurable with respect to a semifinite von Neumann algebra”, Colloq. Math., 55:1 (1988), 109–121 | MR | Zbl

[18] Dodds P.G., Dodds T.K.-Y., de Pagter B., “Noncommutative Köthe duality”, Trans. Amer. Math. Soc., 339:2 (1993), 717–750 | DOI | MR | Zbl

[19] Skvortsova G.Sh., “O slaboi sekventsialnoi polnote faktor-prostranstv prostranstva integriruemykh operatorov”, Izv. vuzov. Matematika, 2002, no. 9, 71–74 | MR | Zbl

[20] Gokhberg I.Ts., Krein M.G., Vvedenie v teoriyu lineinykh nesamosopryazhennykh operatorov, Nauka, M., 1965, 448 pp.

[21] Krein S.G., Petunin Yu.I., Semenov E.M., Interpolyatsiya lineinykh operatorov, Nauka, M., 1978, 400 pp. | MR

[22] Hansen F., “An operator inequality”, Math. Ann., 246:3 (1980), 249–250 | DOI | MR | Zbl

[23] Koliha J.J., “Range projections of idempotents in $C^*$-algebras”, Demonstr. Math., 34:1 (2001), 91–103 | MR | Zbl

[24] Bikchentaev A.M., “O predstavlenii elementov algebry fon Neimana v vide konechnykh summ proizvedenii proektorov”, Sib. mat. zhurn., 46:1 (2005), 32–45 | MR | Zbl

[25] Bikchentaev A.M., “Ob odnom svoistve $L^p$-prostranstv na polukonechnykh algebrakh fon Neimana”, Mat. zametki, 64:2 (1998), 185–190 | MR | Zbl

[26] Householder A.S., Carpenter J.A., “The singular values of involutory and idempotent matrices”, Numer. Math., 5:3 (1963), 234–237 | DOI | MR | Zbl

[27] Strătilă Ş., Zsidó L., Lectures on von Neumann algebras, Abacus Press, Tunbridge Wells (Kent), 1979, 478 pp. | MR | Zbl

[28] Muratov M.A., “Skhodimosti v koltse izmerimykh operatorov”, Sb. nauch. tr. Tash. un-ta, No 573: Funkts. analiz, Izd-vo TashGU, Tashkent, 1978, 51–58 | MR

[29] Rolewicz S., Metric linear spaces, Monogr. mat., 56, PWN, Warszawa, 1972, 287 pp. | MR | Zbl

[30] Banach S., Théorie des opérations linéaires, Monogr. mat., 1, PWN, Warszawa, 1932, 254 pp. | MR

[31] Arens R., “Linear topological division algebras”, Bull. Amer. Math. Soc., 53 (1947), 623–630 | DOI | MR | Zbl

[32] Źelazko W., Metric generalizations of Banach algebras, Rozpr. Mat., 47, PWN, Warszawa, 1965, 70 pp. | MR

[33] Choda H., “An extremal property of the polar decomposition in von Neumann algebras”, Proc. Japan. Acad., 46:4 (1970), 341–344 | DOI | MR | Zbl

[34] Khelemskii A.Ya., Lektsii po funktsionalnomu analizu, Izd-vo MTsNMO, M., 2004, 552 pp.