The Gelfand–Phillips and Dunford–Pettis type properties in bimodules of measurable operators

Huang, Jinghao; Nessipbayev, Yerlan; Pliev, Marat; Sukochev, Fedor

doi:10.1090/tran/9117

The Gelfand–Phillips and Dunford–Pettis type properties in bimodules of measurable operators
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by Jinghao Huang, Yerlan Nessipbayev, Marat Pliev and Fedor Sukochev;

Trans. Amer. Math. Soc. 377 (2024), 6097-6149

DOI: https://doi.org/10.1090/tran/9117

Published electronically: June 21, 2024

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Abstract:

We fully characterize noncommutative symmetric spaces $E(\mathcal {M},\tau )$ affiliated with a semifinite von Neumann algebra $\mathcal {M}$ equipped with a faithful normal semifinite trace $\tau$ on a (not necessarily separable) Hilbert space having the Gelfand–Phillips property and the WCG-property. The complete list of their relations with other classical structural properties (such as the Dunford–Pettis property, the Schur property and their variations) is given in the general setting of noncommutative symmetric spaces.

References

Fernando Albiac and Nigel J. Kalton, Topics in Banach space theory, Graduate Texts in Mathematics, vol. 233, Springer, New York, 2006. MR 2192298
Charalambos D. Aliprantis and Owen Burkinshaw, Positive operators, Pure and Applied Mathematics, vol. 119, Academic Press, Inc., Orlando, FL, 1985. MR 809372
D. Amir and J. Lindenstrauss, The structure of weakly compact sets in Banach spaces, Ann. of Math. (2) 88 (1968), 35–46. MR 228983, DOI 10.2307/1970554
Belmesnaoui Aqzzouz and Khalid Bouras, Weak and almost Dunford-Pettis operators on Banach lattices, Demonstratio Math. 46 (2013), no. 1, 165–179. MR 3075506, DOI 10.1515/dema-2013-0431
Sergey V. Astashkin, Rearrangement invariant spaces satisfying Dunford-Pettis criterion of weak compactness, Functional analysis and geometry: Selim Grigorievich Krein centennial, Contemp. Math., vol. 733, Amer. Math. Soc., [Providence], RI, [2019] ©2019, pp. 45–59. MR 3985266, DOI 10.1090/conm/733/14732
Sergey V. Astashkin, Nigel Kalton, and Fyodor A. Sukochev, Cesaro mean convergence of martingale differences in rearrangement invariant spaces, Positivity 12 (2008), no. 3, 387–406. MR 2421142, DOI 10.1007/s11117-007-2146-y
S. V. Astashkin and F. A. Sukochev, Banach-Saks property in Marcinkiewicz spaces, J. Math. Anal. Appl. 336 (2007), no. 2, 1231–1258. MR 2353012, DOI 10.1016/j.jmaa.2007.03.040
Jonathan Arazy, Basic sequences, embeddings, and the uniqueness of the symmetric structure in unitary matrix spaces, J. Functional Analysis 40 (1981), no. 3, 302–340. MR 611587, DOI 10.1016/0022-1236(81)90052-5
Colin Bennett and Robert Sharpley, Interpolation of operators, Pure and Applied Mathematics, vol. 129, Academic Press, Inc., Boston, MA, 1988. MR 928802
J. Borwein, M. Fabian, and J. Vanderwerff, Characterizations of Banach spaces via convex and other locally Lipschitz functions, Acta Math. Vietnam. 22 (1997), no. 1, 53–69. MR 1479738
A. V. Buhvalov, Locally convex spaces that are generated by weakly compact sets, Vestnik Leningrad. Univ. 7, Mat. Meh. Astronom. Vyp. 2 (1973), 11–17, 160 (Russian, with English summary). MR 361701
A. V. Buhvalov, A. I. Veksler, and V. A. Geĭler, Normed lattices, Mathematical analysis, Vol. 18 (Russian), Itogi Nauki i Tehniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1980, pp. 125–184 (Russian). MR 597904
A. V. Buhvalov, A. I. Veksler, and G. Ja. Lozanovskiĭ, Banach lattices — some Banach aspects of the theory, Uspekhi Mat. Nauk 34 (1979), no. 2(206), 137–183 (Russian). MR 535711
Jean Bourgain and Joe Diestel, Limited operators and strict cosingularity, Math. Nachr. 119 (1984), 55–58. MR 774176, DOI 10.1002/mana.19841190105
L. J. Bunce, The Dunford-Pettis property in the predual of a von Neumann algebra, Proc. Amer. Math. Soc. 116 (1992), no. 1, 99–100. MR 1091177, DOI 10.1090/S0002-9939-1992-1091177-1
H. Carrión, P. Galindo, and M. Lourenço, A stronger Dunford-Pettis property, Studia Math. 184 (2008), no. 3, 205–216. MR 2369139, DOI 10.4064/sm184-3-1
J. M. F. Castillo and M. Gonzáles, On the Dunford-Pettis property in Banach spaces, Acta Univ. Carolin. Math. Phys. 35 (1994), no. 2, 5–12. MR 1366492
Jin Xi Chen, Zi Li Chen, and Guo Xing Ji, Almost limited sets in Banach lattices, J. Math. Anal. Appl. 412 (2014), no. 1, 547–553. MR 3145821, DOI 10.1016/j.jmaa.2013.10.085
Jin Xi Chen, Zi Li Chen, and Guo Xing Ji, Domination by positive weak* Dunford-Pettis operators on Banach lattices, Bull. Aust. Math. Soc. 90 (2014), no. 2, 311–318. MR 3252014, DOI 10.1017/S000497271400032X
V. I. Chilin, P. G. Dodds, and F. A. Sukochev, The Kadec-Klee property in symmetric spaces of measurable operators, Israel J. Math. 97 (1997), 203–219. MR 1441249, DOI 10.1007/BF02774037
Vladimir I. Chilin, Andrei V. Krygin, and Pheodor A. Sukochev, Local uniform and uniform convexity of noncommutative symmetric spaces of measurable operators, Math. Proc. Cambridge Philos. Soc. 111 (1992), no. 2, 355–368. MR 1142755, DOI 10.1017/S0305004100075459
V. I. Chilin and F. A. Sukochev, Weak convergence in non-commutative symmetric spaces, J. Operator Theory 31 (1994), no. 1, 35–65. MR 1316983
Cho-Ho Chu and Bruno Iochum, The Dunford-Pettis property in $C^*$-algebras, Studia Math. 97 (1990), no. 1, 59–64. MR 1074769, DOI 10.4064/sm-97-1-59-64
John B. Conway, A course in functional analysis, Graduate Texts in Mathematics, vol. 96, Springer-Verlag, New York, 1985. MR 768926, DOI 10.1007/978-1-4757-3828-5
Małgorzata Czerwińska and Anna Kamińska, Geometric properties of noncommutative symmetric spaces of measurable operators and unitary matrix ideals, Comment. Math. 57 (2017), no. 1, 45–122. MR 3703594, DOI 10.14708/cm.v57i1.3291
Joseph Diestel, Geometry of Banach spaces—selected topics, Lecture Notes in Mathematics, Vol. 485, Springer-Verlag, Berlin-New York, 1975. MR 461094, DOI 10.1007/BFb0082079
Joe Diestel, A survey of results related to the Dunford-Pettis property, Proceedings of the Conference on Integration, Topology, and Geometry in Linear Spaces (Univ. North Carolina, Chapel Hill, N.C., 1979) Contemp. Math., vol. 2, Amer. Math. Soc., Providence, RI, 1980, pp. 15–60. MR 621850
Joseph Diestel, Sequences and series in Banach spaces, Graduate Texts in Mathematics, vol. 92, Springer-Verlag, New York, 1984. MR 737004, DOI 10.1007/978-1-4612-5200-9
J. Diestel and J. J. Uhl Jr., Vector measures, Mathematical Surveys, No. 15, American Mathematical Society, Providence, RI, 1977. With a foreword by B. J. Pettis. MR 453964, DOI 10.1090/surv/015
Sjoerd Dirksen, Ben de Pagter, Denis Potapov, and Fedor Sukochev, Rosenthal inequalities in noncommutative symmetric spaces, J. Funct. Anal. 261 (2011), no. 10, 2890–2925. MR 2832586, DOI 10.1016/j.jfa.2011.07.015
Ivan Dobrakov, On representation of linear operators on $C_{0}\,(T,\,{\rm {\bf }X})$, Czechoslovak Math. J. 21(96) (1971), 13–30. MR 276804, DOI 10.21136/CMJ.1971.101000
Peter G. Dodds, Theresa K. Dodds, and Ben de Pagter, Weakly compact subsets of symmetric operator spaces, Math. Proc. Cambridge Philos. Soc. 110 (1991), no. 1, 169–182. MR 1104612, DOI 10.1017/S0305004100070225
Peter G. Dodds, Theresa K.-Y. Dodds, and Ben de Pagter, Noncommutative Köthe duality, Trans. Amer. Math. Soc. 339 (1993), no. 2, 717–750. MR 1113694, DOI 10.1090/S0002-9947-1993-1113694-3
P. G. Dodds and B. de Pagter, Properties $(\textrm {u})$ and $(\textrm {V}^*)$ of Pelczynski in symmetric spaces of $\tau$-measurable operators, Positivity 15 (2011), no. 4, 571–594. MR 2861600, DOI 10.1007/s11117-011-0127-7
P. G. Dodds and B. de Pagter, The non-commutative Yosida-Hewitt decomposition revisited, Trans. Amer. Math. Soc. 364 (2012), no. 12, 6425–6457. MR 2958942, DOI 10.1090/S0002-9947-2012-05569-3
P. G. Dodds and B. de Pagter, Normed Köthe spaces: a non-commutative viewpoint, Indag. Math. (N.S.) 25 (2014), no. 2, 206–249. MR 3151815, DOI 10.1016/j.indag.2013.01.009
P. G. Dodds, B. de Pagter, and F. Sukochev, Sets of uniformly absolutely continuous norm in symmetric spaces of measurable operators, Trans. Amer. Math. Soc. 368 (2016), no. 6, 4315–4355. MR 3453373, DOI 10.1090/tran/6477
P. Dodds, B. de Pagter, and F. Sukochev, Noncommutative integration and operator theory, Birkhäuser Cham, Progress in Mathematics, vol. 349, Springer Nature Switzerland AG, 2023, https://link.springer.com/book/9783031496530.
P. G. Dodds, F. A. Sukochev, and G. Schlüchtermann, Weak compactness criteria in symmetric spaces of measurable operators, Math. Proc. Cambridge Philos. Soc. 131 (2001), no. 2, 363–384. MR 1857125, DOI 10.1017/S0305004101005114
Lech Drewnowski, On Banach spaces with the Gel′fand-Phillips property, Math. Z. 193 (1986), no. 3, 405–411. MR 862887, DOI 10.1007/BF01229808
Nelson Dunford and B. J. Pettis, Linear operations on summable functions, Trans. Amer. Math. Soc. 47 (1940), 323–392. MR 2020, DOI 10.1090/S0002-9947-1940-0002020-4
G. Emmanuele, Gel′fand-Phillips property in a Banach space of vector valued measures, Math. Nachr. 127 (1986), 21–23. MR 861715, DOI 10.1002/mana.19861270103
M. Fabian, G. Godefroy, V. Montesinos, and V. Zizler, Inner characterizations of weakly compactly generated Banach spaces and their relatives, J. Math. Anal. Appl. 297 (2004), no. 2, 419–455. Special issue dedicated to John Horváth. MR 2088670, DOI 10.1016/j.jmaa.2004.02.015
Marián Fabian, Petr Habala, Petr Hájek, Vicente Montesinos Santalucía, Jan Pelant, and Václav Zizler, Functional analysis and infinite-dimensional geometry, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 8, Springer-Verlag, New York, 2001. MR 1831176, DOI 10.1007/978-1-4757-3480-5
Thierry Fack and Hideki Kosaki, Generalized $s$-numbers of $\tau$-measurable operators, Pacific J. Math. 123 (1986), no. 2, 269–300. MR 840845, DOI 10.2140/pjm.1986.123.269
A. Grothendieck, Sur les applications linéaires faiblement compactes d’espaces du type $C(K)$, Canad. J. Math. 5 (1953), 129–173 (French). MR 58866, DOI 10.4153/cjm-1953-017-4
Jinghao Huang, Marat Pliev, and Fedor Sukochev, (Non-)Dunford–Pettis operators on noncommutative symmetric spaces, J. Funct. Anal. 282 (2022), no. 11, Paper No. 109443, 29. MR 4392281, DOI 10.1016/j.jfa.2022.109443
Jinghao Huang and Fedor Sukochev, Isomorphic classification of $L_{p, q}$-spaces, II, J. Funct. Anal. 280 (2021), no. 12, Paper No. 108994, 34. MR 4234225, DOI 10.1016/j.jfa.2021.108994
J. Huang and F. Sukochev, Derivations with values in noncommutative symmetric spaces, C. R. Math. Acad. Sci. Paris 361 (2023), 1357–1365.
J. Huang and F. Sukochev, For which noncommutative function spaces must a derivation be inner?, preprint.
Jesús Angel Jaramillo, Angeles Prieto, and Ignacio Zalduendo, Sequential convergences and Dunford-Pettis properties, Ann. Acad. Sci. Fenn. Math. 25 (2000), no. 2, 467–475. MR 1762430
Ep de Jonge, The semi-$M$ property for normed Riesz spaces, Compositio Math. 34 (1977), no. 2, 147–172. MR 448024
Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Vol. I, Pure and Applied Mathematics, vol. 100, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. Elementary theory. MR 719020
N. J. Kalton and F. A. Sukochev, Symmetric norms and spaces of operators, J. Reine Angew. Math. 621 (2008), 81–121. MR 2431251, DOI 10.1515/CRELLE.2008.059
Anna Kamińska and Mieczysław Mastyło, The Dunford-Pettis property for symmetric spaces, Canad. J. Math. 52 (2000), no. 4, 789–803. MR 1767402, DOI 10.4153/CJM-2000-033-9
Anna Kamińska and Mieczysław Mastyło, The Schur and (weak) Dunford-Pettis properties in Banach lattices, J. Aust. Math. Soc. 73 (2002), no. 2, 251–278. MR 1926073, DOI 10.1017/S144678870000882X
S. G. Kreĭn, Yu. Ī. Petunīn, and E. M. Semënov, Interpolation of linear operators, Translations of Mathematical Monographs, vol. 54, American Mathematical Society, Providence, RI, 1982. Translated from the Russian by J. Szűcs. MR 649411
Denny H. Leung, On the weak Dunford-Pettis property, Arch. Math. (Basel) 52 (1989), no. 4, 363–364. MR 998412, DOI 10.1007/BF01194411
Pei-Kee Lin, Köthe-Bochner function spaces, Birkhäuser Boston, Inc., Boston, MA, 2004. MR 2018062, DOI 10.1007/978-0-8176-8188-3
Joram Lindenstrauss, Weakly compact sets—their topological properties and the Banach spaces they generate, Symposium on Infinite-Dimensional Topology (Louisiana State Univ., Baton Rouge, La., 1967) Ann. of Math. Stud., No. 69, Princeton Univ. Press, Princeton, NJ, 1972, pp. 235–273. MR 417761
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Springer-Verlag, Berlin-New York, 1979. Function spaces. MR 540367, DOI 10.1007/978-3-662-35347-9
Steven Lord, Fedor Sukochev, and Dmitriy Zanin, Singular traces, De Gruyter Studies in Mathematics, vol. 46, De Gruyter, Berlin, 2013. Theory and applications. MR 3099777
A. C. Zaanen, Riesz spaces. II, North-Holland Mathematical Library, vol. 30, North-Holland Publishing Co., Amsterdam, 1983. MR 704021, DOI 10.1016/S0924-6509(08)70234-4
G. Ya. Lozanovskiĭ, Mappings of Banach lattices of measurable functions, Izv. Vyssh. Uchebn. Zaved. Mat. 5(192) (1978), 84–86 (Russian). MR 509135
A. M. Medzhitov, Separability of noncommutative symmetric spaces, Mathematical analysis and algebra (Russian), Tashkent. Gos. Univ., Tashkent, 1986, pp. 38–43, 97 (Russian). MR 1061594
Peter Meyer-Nieberg, Banach lattices, Universitext, Springer-Verlag, Berlin, 1991. MR 1128093, DOI 10.1007/978-3-642-76724-1
Edward Nelson, Notes on non-commutative integration, J. Functional Analysis 15 (1974), 103–116. MR 355628, DOI 10.1016/0022-1236(74)90014-7
Timur Oikhberg and Eugeniu Spinu, Domination of operators in the non-commutative setting, Studia Math. 219 (2013), no. 1, 35–67. MR 3139423, DOI 10.4064/sm219-1-3
Prakash Pethe and Nimbakrishna Thakare, Note on Dunford-Pettis property and Schur property, Indiana Univ. Math. J. 27 (1978), no. 1, 91–92. MR 458123, DOI 10.1512/iumj.1978.27.27008
H. Pfitzner, Weak compactness in the dual of a $C^\ast$-algebra is determined commutatively, Math. Ann. 298 (1994), no. 2, 349–371. MR 1256621, DOI 10.1007/BF01459739
Frank Räbiger, Beiträge zur Strukturtheorie der Grothendieck-Räume, Sitzungsberichte der Heidelberger Akademie der Wissenschaften. Mathematisch-Naturwissenschaftliche Klasse [Reports of the Heidelberg Academy of Science. Section for Mathematics and Natural Sciences], vol. 85, Springer-Verlag, Berlin, 1985 (German). MR 828457, DOI 10.1007/978-3-642-45612-1
Abderrahman Retbi, On the class of positive almost weak$^\star$ Dunford-Pettis operators, Comment. Math. Univ. Carolin. 56 (2015), no. 3, 347–354. MR 3390281, DOI 10.14712/1213-7243.2015.128
Haskell P. Rosenthal, The heredity problem for weakly compactly generated Banach spaces, Compositio Math. 28 (1974), 83–111. MR 417762
J. A. Sanchez, Operators on Banach lattices, Ph.D. Thesis, Complutense University, Madrid, 1985 (in Spanish).
Martin Schechter, Principles of functional analysis, 2nd ed., Graduate Studies in Mathematics, vol. 36, American Mathematical Society, Providence, RI, 2002. MR 1861991, DOI 10.1090/gsm/036
T. Schlumprecht, Limited sets in Banach spaces, Ph.D. Thesis, München, 1987.
J. Schur, Über lineare Transformationen in der Theorie der unendlichen Reihen, J. Reine Angew. Math. 151 (1921), 79–111 (German). MR 1580985, DOI 10.1515/crll.1921.151.79
E. M. Semenov and F. A. Sukochev, Sums and intersections of symmetric operator spaces, J. Math. Anal. Appl. 414 (2014), no. 2, 742–755. MR 3167994, DOI 10.1016/j.jmaa.2013.12.039
F. Sukochev, Symmetric spaces of measurable operators on finite von Neumann algebras, Ph.D. Thesis, Tashkent State University, 1988.
F. A. Sukochev, Linear-topological classification of separable $L_p$-spaces associated with von Neumann algebras of type I, Israel J. Math. 115 (2000), 137–156. MR 1749676, DOI 10.1007/BF02810584
Masamichi Takesaki, Theory of operator algebras. I, Springer-Verlag, New York-Heidelberg, 1979. MR 548728, DOI 10.1007/978-1-4612-6188-9
Witold Wnuk, Banach lattices with the weak Dunford-Pettis property, Atti Sem. Mat. Fis. Univ. Modena 42 (1994), no. 1, 227–236. MR 1282338
W. Wnuk, Banach lattices with order continuous norms, Advanced Topics in Mathematics, Adam Mickiewicz Univ., Poznan, Polish Scientific Publishers PWN, 1999.

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The Gelfand–Phillips and Dunford–Pettis type properties in bimodules of measurable operators
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