Topological algebras of measurable and locally measurable operators

M. A. Muratov ; V. I. Chilin

Contemporary Mathematics. Fundamental Directions, Proceedings of the Crimean autumn mathematical school-symposium, Tome 61 (2016), pp. 115-163.

Résumé

In this paper, we review the results on topological $*$-algebras $S(\mathcal M)$, $S(\mathcal M,\tau)$ and $LS(\mathcal M)$ of measurable, $\tau$-measurable, and locally measurable operators affiliated with the von Neumann algebra $\mathcal M$. Also we consider relations between these algebras for different classes of von Neumann algebras and establish the continuity of operator-valued functions with respect to local convergence in measure. We describe maximal commutative $*$-subalgebras of the algebra $LS(\mathcal M)$ as well.

Bibliographie

[1] A. M. Bikchentaev, “Local convergence in measure on semifinite von Neumann algebras”, Proc. Math. Inst. Russ. Acad. Sci., 255, 2006, 41–54 (in Russian) | MR | Zbl

[2] A. M. Bikchentaev, “Local convergence in measure on semifinite von Neumann algebras. II”, Math. Notes, 82:5 (2007), 783–786 (in Russian) | DOI | MR | Zbl

[3] U. Bratelli, D. Robinson, Operator Algebras and Quantum Statistical Mechanics, Mir, Moscow, 1982 (in Russian) | MR

[4] I. M. Gel'fand, M. A. Naymark, “On the inclusion of the normed ring into the ring of operators in a Hilbert space”, Math. Digest, 12(54):2 (1943), 197–217 (in Russian) | MR | Zbl

[5] B. S. Zakirov, V. I. Chilin, “Abstract characterization of $EW^*$-algebras”, Funct. Anal. Appl., 25:1 (1991), 76–78 (in Russian) | MR | Zbl

[6] B. S. Zakirov, V. I. Chilin, “Description of $GB^*$-algebras that have a $W^*$ algebra as a bounded part”, Uzbek Math. J., 1991, no. 2, 24–29 (in Russian) | MR

[7] M. A. Muratov, V. I. Chilin, “$*$-Algebras of measurable and locally measurable operators affiliated with the von Neumann algebra”, Rep. Ukrainian Nat. Acad. Sci., 2005, no. 9, 28–30 (in Russian) | MR | Zbl

[8] M. A. Muratov, B. I. Chilin, “$*$-Algebras of unbounded operators affiliated with the von Neumann algebra”, Notes Sci. Semin. Saint-Petersburg Dept. Math. Inst. Russ. Acad. Sci., 326, 2005, 183–197 (in Russian) | MR | Zbl

[9] M. A. Muratov, V. I. Chilin, Algebras of Measurable and Locally Measurable Operators, Pratsi In-t. Matem. NAN Ukraïni, Kiïv, 2007 (in Russian)

[10] M. A. Muratov, V. I. Chilin, “Central extensions of $*$-algebra of bounded operators”, Rep. Ukrainian Nat. Acad. Sci., 2009, no. 7, 24–28 (in Russian) | Zbl

[11] M. A. Naymark, Normed Rings, Nauka, Moscow, 1968 (in Russian) | MR

[12] F. Riess, B. Szokefalvi-Nagy, Lectures in Functional Analysis, Mir, Moscow, 1979 (in Russian) | MR

[13] W. Rudin, Functional Analysis, Mir, Moscow, 1975 (in Russian) | MR

[14] T. A. Sarymsakov, Sh. A. Ayupov, D. Khadzhiev, V. I. Chilin, Ordered Algebras, FAN, Tashkent, 1983 (in Russian) | MR

[15] F. A. Sukochev, V. I. Chilin, “The triangle inequality for measurable operators with respect to the Hardy–Littlewood ordering”, Bull. Acad. Sci. Uzbek SSR. Ser. Phys. Math. Sci., 1988, no. 4, 44–50 (in Russian) | MR | Zbl

[16] O. E. Tikhonov, “Continuity of operator functions in topologies connected to the trace on the Neumann algebra”, Bull. Higher Edu. Inst. Ser. Math., 1987, no. 1, 77–79 | MR | Zbl

[17] Antoine J. P., Inoue A., Trapani C., Partial $*$-algebras and their operator realizations, Kluwer Academic Publishers, Dordrecht, 2002 | MR | Zbl

[18] Ber A. F., Chilin V. I., Sukochev F. A., “Continuous derivations on algebras of locally measurable operators are inner”, Proc. Lond. Math. Soc., 109:3 (2014), 65–89 | DOI | MR | Zbl

[19] Chilin V. I., Muratov M. A., “Comparison of topologies on $*$-algebras of locally measurable operators”, Positivity, 17:1 (2013), 111–132 | DOI | MR | Zbl

[20] Chilin V. I., Muratov M. A., “Continuity of operator-valued functions in the $*$-algebra of locally measurable operators”, Methods Funct. Anal. Topology, 20:2 (2014), 124–134 | MR

[21] Davies E. B., “A generalization of Kaplansky's theory”, J. Lond. Math. Soc., 4 (1972), 435-436 | DOI | Zbl

[22] Dieudonne J., Foundations of modern analysis, Acad. Press, New York–London, 1960 | MR | Zbl

[23] Dixmier J., Les Algebres d'operateurs dans l'espace Hilbertien (Algebres de von Neumann), Gauthier-Villars, Paris, 1969 | MR | Zbl

[24] Dixon P. G., “Unboubded operator algebras”, Proc. Lond. Math. Soc., 23:3 (1973), 53–59 | MR

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

[26] Kadison R. V., “Strong continuity of operator functions”, Pacific J. Math., 26 (1968), 121–129 | DOI | MR | Zbl

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

[28] Kaplansky I., “A theorem on rings operators”, Pacific J. Math., 1 (1951), 227–232 | DOI | MR | Zbl

[29] Kunce R. A., “$L^p$ Fourier transforms on locally compact unimodular groups”, Trans. Am. Math. Soc., 89 (1958), 519–540 | MR

[30] Muratov M. A., Chilin V. I., “$*$-Algebras of unbounded operators affiliated with a von Neumann algebra”, J. Math. Sci. (N.Y.), 140:3 (2007), 445–451 | DOI | MR | Zbl

[31] Murphy G. J., $C^*$-algebras and operator theory, Academic Press, Inc., New York–London, 1990 | MR

[32] Murrey F. J., von Neumann J., “On ring of operators”, Ann. Math., 37 (1936), 116–229 | DOI

[33] Murrey F. J., von Neumann J., “On ring of operators. II”, Trans. Am. Math. Soc., 41 (1937), 208–248 | DOI | MR

[34] Murrey F. J., von Neumann J., “On ring of operators. IV”, Ann. Math., 44 (1943), 716–808 | DOI | MR

[35] Nelson E., “Notes on noncommutative integration”, J. Funct. Anal., 15 (1974), 103–116 | DOI | MR | Zbl

[36] von Neumann J., “On ring of operators. III”, Ann. Math., 41 (1940), 94–161 | DOI | MR | Zbl

[37] Padmanabhan A. R., “Convergence in measure and related results in finite rings of operators”, Trans. Am. Math. Soc., 128 (1967), 359–388 | DOI | MR

[38] Reed M., Simon B., Methods of modern mathematical physics, v. I, Functional Analysis, Academic Press, New York, 1980 | MR | Zbl

[39] Sakai S., $C^*$-algebras and $W^*$-algebras, Springer, New York, 1971 | MR

[40] Sankaran S., “The $*$-algebra of unbounded operators”, J. Lond. Math. Soc., 343 (1959), 337–344 | DOI | MR

[41] Sankaran S., “Stochastic convergence for operators”, Quart. J. Math., 2:15 (1964), 97–102 | DOI | MR

[42] Schmudgen K., Unbounded operator algebras an representation theory, Birkhauser, Basel, 1990 | MR

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

[44] Stinespring W. E., “Integration theorems for gages and duality for unimodular groups”, Trans. Am. Math. Soc., 90 (1959), 15–56 | DOI | MR | Zbl

[45] Strătilă S., Zsidó L., Lectures on von Neumann algebras, Abacus Press, Bucharest, 1979 | MR | Zbl

[46] Takesaki M., Theory of operator algebras, v. I, Springer, New York, 1979 | MR | Zbl

[47] Yeadon F. J., “Convergence of measurable operators”, Proc. Camb. Philos. Soc., 74 (1973), 257–268 | DOI | MR | Zbl

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