×

Topological algebras of measurable and locally measurable operators. (English. Russian original) Zbl 1428.46038

J. Math. Sci., New York 239, No. 5, 654-705 (2019); translation from Sovrem. Mat., Fundam. Napravl. 61, 115-163 (2016).
This is a survey of results on \(^*\)-algebras of measurable, \(\tau\)-measurable, locally measurable and \(\tau\)-locally measurable operators affiliated with a von Neumann algebra, with some new results added. The survey starts with an introduction to the theory of von Neumann algebras, which makes the material accessible to a graduate student. In the next part, the algebras are defined and their basic properties enumerated. Special attention is given to the order properties. Then the different algebras of measurable operators are compared, and the dependence of \(\tau\)-measurability on \(\tau\) is investigated. Finally, the authors look at the corresponding measure topologies on the algebras and prove theorems on the continuity of functions defined on and taking values in the algebra of locally measurable operators. Most of the results presented have been obtained in earlier papers of the authors.

MSC:

46L51 Noncommutative measure and integration
46H35 Topological algebras of operators
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] J.P. Antoine, A. Inoue, and C. Trapani, Partial ∗-Algebras and Their Operator Realizations, Kluwer Academic Publishers, Dordrecht (2002). · Zbl 1023.46004
[2] A. F. Ber, V. I. Chilin, and F. A. Sukochev, “Continuous derivations on algebras of locally measurable operators are inner,” Proc. London Math. Soc., 109, No. 3, 65-89 (2014). · Zbl 1304.46064 · doi:10.1112/plms/pdt070
[3] A. M. Bikchentaev, “Local convergence in measure on semifinite von Neumann algebras,” Tr. Mat. Inst. Steklova, 255, 41-54 (2006). · Zbl 1351.46057
[4] A. M. Bikchentaev, “Local convergence in measure on semifinite von Neumann algebras. II,” Mat. Zametki, 82, No. 5, 703-707 (2007). · Zbl 1152.46313
[5] U. Bratelli and D. Robinson, Operator Algebras and Quantum Statistical Mechanics [Russian translation], Mir, Moscow (1982). · Zbl 0523.46043
[6] V. I. Chilin and M.A. Muratov, “Comparison of topologies on <Emphasis Type=”Italic“>∗-algebras of locally measurable operators,” Positivity, 17, No. 1, 111-132 (2013). · Zbl 1331.46049
[7] V. I. Chilin and M.A. Muratov, “Continuity of operator-valued functions in the <Emphasis Type=”Italic“>∗-algebra of locally measurable operators,” Methods Funct. Anal. Topology, 20, No. 2, 124-134 (2014). · Zbl 1313.46066
[8] E. B. Davies, “A generalization of Kaplansky’s theory,” J. London Math. Soc., 4, 435-436 (1972). · Zbl 0229.47015 · doi:10.1112/jlms/s2-4.3.435
[9] J. Dieudonne, Foundations of Modern Analysis, Acad. Press, New York-London (1960). · Zbl 0100.04201
[10] J. Dixmier, Les Algebres d’Operateurs dans l’Espace Hilbertien (Algebres de von Neumann), Gauthier-Villars, Paris (1969). · Zbl 0175.43801
[11] P.G. Dixon, “Unbounded operator algebras,” Proc. London Math. Soc., 23, No. 3, 53-59 (1973). · Zbl 0216.41501
[12] T. Fack and H. Kosaki, “Generalized <Emphasis Type=”Italic“>s-numbers of <Emphasis Type=”Italic“>τ -measurable operators,” Pacific J. Math., 123, 269-300 (1986). · Zbl 0617.46063 · doi:10.2140/pjm.1986.123.269
[13] I. M. Gel’fand and M. A. Naymark, “On the inclusion of the normed ring into the ring of operators in a Hilbert space,” Mat. Sb., 12, 197-213 (1943). · Zbl 0060.27006
[14] R. V. Kadison, “Strong continuity of operator functions,” Pacific J. Math., 26, 121-129 (1968). · Zbl 0169.16902 · doi:10.2140/pjm.1968.26.121
[15] I. Kaplansky, “Projections in Banach algebras,” Ann. of Math., 53, 235-249 (1951). · Zbl 0042.12402 · doi:10.2307/1969540
[16] I. Kaplansky, “A theorem on rings operators,” Pacific J. Math., 1, 227-232 (1951). · Zbl 0043.11502 · doi:10.2140/pjm.1951.1.227
[17] R. A. Kunce, “<Emphasis Type=”Italic“>L <Emphasis Type=”Italic“>p Fourier transforms on locally compact unimodular groups,” Trans. Amer. Math. Soc., 89, 519-540 (1958). · Zbl 0084.33905
[18] M. A. Muratov and V. I. Chilin, “<Emphasis Type=”Italic“>∗-Algebras of measurable and locally measurable operators affiliated with the von Neumann algebra,” Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki, 9, 28-30 (2005). · Zbl 1095.46511
[19] M. A. Muratov and B. I. Chilin, “<Emphasis Type=”Italic“>∗-Algebras of unbounded operators affiliated with the von Neumann algebra,” Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 326, 183-197 (2005). · Zbl 1098.47034
[20] M. A. Muratov and V. I. Chilin, Algebras of Measurable and Locally Measurable Operators [in Russian], Pratsi In-t. Matem. NAN Ukraïni, Kiïv (2007). · Zbl 1199.47002
[21] M. A. Muratov and V. I. Chilin, “<Emphasis Type=”Italic“>∗-Algebras of unbounded operators affiliated with a von Neumann algebra,” J. Math. Sci. (N.Y.), 140, No. 3, 445-451 (2007).
[22] M. A. Muratov and V. I. Chilin, “Central extensions of <Emphasis Type=”Italic“>∗-algebra of bounded operators,” Dopov. Nats. Akad. Nauk Ukr., Mat. Prirodozn. Tekh. Nauky, 7, 24-28 (2009). · Zbl 1199.46136
[23] G. J. Murphy, C⋆-Algebras and Operator Theory, Academic Press, Inc., New York-London (1990). · Zbl 0714.46041
[24] F. J. Murrey and J. von Neumann, “On ring of operators,” Ann. of Math., 37, 116-229 (1936).
[25] F. J. Murrey and J. von Neumann, “On ring of operators. II,” Trans. Amer. Math. Soc., 41, 208-248 (1937). · Zbl 0017.36001
[26] F. J. Murrey and J. von Neumann, “On ring of operators. IV,” Ann. of Math., 44, 716-808 (1943). · Zbl 0060.26903 · doi:10.2307/1969107
[27] M. A. Naymark, Normed Rings [in Russian], Nauka, Moscow (1968).
[28] E. Nelson, “Notes on noncommutative integration,” J. Funct. Anal., 15, 103-116 (1974). · Zbl 0292.46030 · doi:10.1016/0022-1236(74)90014-7
[29] J. von Neumann, “On ring of operators. III,” Ann. of Math., 41, 94-161 (1940). · doi:10.2307/1968823
[30] A. R. Padmanabhan, “Convergence in measure and related results in finite rings of operators,” Trans. Amer. Math. Soc., 128, 359-388 (1967). · Zbl 0166.11503 · doi:10.1090/S0002-9947-1967-0212581-7
[31] M. Reed and B. Simon, Methods of Modern Mathematical Physics. I. Functional Analysis, Academic Press, New York (1980). · Zbl 0459.46001
[32] F. Riesz and B. Sz.-Nagy, Lectures in Functional Analysis [Russian translation], Mir, Moscow (1979). · Zbl 0046.33103
[33] W. Rudin, Functional Analysis [Russian translation], Mir, Moscow (1975). · Zbl 0253.46001
[34] S. Sakai, C⋆-Algebras and W⋆-Algebras, Springer, New York (1971). · Zbl 0219.46042
[35] S. Sankaran, “The <Emphasis Type=”Italic“>∗-algebra of unbounded operators,” J. London Math. Soc., 343, 337-344 (1959). · Zbl 0088.32501
[36] S. Sankaran, “Stochastic convergence for operators,” Q. J. Math., 2, No. 15, 97-102 (1964). · Zbl 0138.38503
[37] T. A. Sarymsakov, Sh. A. Ayupov, D. Khadzhiev, and V. I. Chilin, Ordered Algebras [in Russian], FAN, Tashkent (1983). · Zbl 0542.46001
[38] K. Schmudgen, Unbounded Operator Algebras an Representation Theory, Birkhäuser, Basel (1990). · Zbl 0697.47048
[39] I.E. Segal, “A non-commutative extension of abstract integration,” Ann. of Math., 57, 401-457 (1953). · Zbl 0051.34201 · doi:10.2307/1969729
[40] W. E. Stinespring, “Integration theorems for gages and duality for unimodular groups,” Trans. Amer. Math. Soc., 90, 15-56 (1959). · Zbl 0085.10202 · doi:10.1090/S0002-9947-1959-0102761-9
[41] S. Strătilă and L. Zsidó, Lectures on von Neumann Algebras, Abacus Press, Bucharest (1979). · Zbl 0391.46048
[42] F. A. Sukochev and V. I. Chilin, “The triangle inequality for measurable operators with respect to the Hardy-Littlewood ordering,” Izv. Akad. Nauk UzSSR, Ser. Fiz.-Mat. Nauk, 4, 44-50 (1988). · Zbl 0673.46044
[43] M. Takesaki, Theory of Operator Algebras. I, Springer, New York (1979). · Zbl 0436.46043
[44] O.E. Tikhonov, “Continuity of operator functions in topologies connected to the trace on the Neumann algebra,” Izv. Vyssh. Uchebn. Zaved. Mat., 1, 77-79 (1987).
[45] F. J. Yeadon, “Convergence of measurable operators,” Math. Proc. Cambridge Philos. Soc., 74, 257-268 (1973). · Zbl 0272.46043 · doi:10.1017/S0305004100048052
[46] F. J. Yeadon, “Non-commutative <Emphasis Type=”Italic“>L <Emphasis Type=”Italic“>p-spaces,” Math. Proc. Cambridge Philos. Soc., 77, 91-102 (1975). · Zbl 0327.46068 · doi:10.1017/S0305004100049434
[47] B. S. Zakirov and V. I. Chilin, “Abstract characterization of <Emphasis Type=”Italic“>EW <Emphasis Type=”Italic“>∗-algebras,” Funktsional. Anal. i Prilozhen., 25, No. 1, 76-78 (1991). · Zbl 0826.47032
[48] B. S. Zakirov and V. I. Chilin, “Description of <Emphasis Type=”Italic“>GB <Emphasis Type=”Italic“>∗-algebras that have a <Emphasis Type=”Italic“>W <Emphasis Type=”Italic“>∗ algebra as a bounded part,” Uzbek. Mat. Zh., No. 2, 24-29 (1991).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.