Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Sets of uniformly absolutely continuous norm in symmetric spaces of measurable operators


Authors: P. G. Dodds, B. de Pagter and F. Sukochev
Journal: Trans. Amer. Math. Soc. 368 (2016), 4315-4355
MSC (2010): Primary 46L52; Secondary 46E30, 47A30
DOI: https://doi.org/10.1090/tran/6477
Published electronically: September 15, 2015
MathSciNet review: 3453373
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We characterise sets of uniformly absolutely continuous norm in strongly symmetric spaces of $ \tau $-measurable operators. Applications are given to the study of relatively weakly compact and relatively compact sets and to compactness properties of operators dominated in the sense of complete positivity by compact or by Dunford-Pettis operators.


References [Enhancements On Off] (What's this?)

  • [1] Charalambos D. Aliprantis and Owen Burkinshaw, Positive operators, Pure and Applied Mathematics, vol. 119, Academic Press, Inc., Orlando, FL, 1985. MR 809372 (87h:47086)
  • [2] Charles A. Akemann, The dual space of an operator algebra, Trans. Amer. Math. Soc. 126 (1967), 286-302. MR 0206732 (34 #6549)
  • [3] 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 (2010k:46015), https://doi.org/10.1007/s11117-007-2146-y
  • [4] T. Andô, On fundamental properties of a Banach space with a cone, Pacific J. Math. 12 (1962), 1163-1169. MR 0150572 (27 #568)
  • [5] A. M. Bikchentaev, A block projection operator in normed ideal spaces of measurable operators, Izv. Vyssh. Uchebn. Zaved. Mat. 2 (2012), 86-91 (Russian, with English and Russian summaries); English transl., Russian Math. (Iz. VUZ) 56 (2012), no. 2, 75-79. MR 3076533, https://doi.org/10.3103/S1066369X12020107
  • [6] Vladimir I. Chilin, Andrei V. Krygin, and Pheodor A. Sukochev, Extreme points of convex fully symmetric sets of measurable operators, Integral Equations Operator Theory 15 (1992), no. 2, 186-226. MR 1147280 (93g:46065), https://doi.org/10.1007/BF01204237
  • [7] 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 (96e:46085)
  • [8] V. I. Chilin, A. A. Sedaev, and F. A. Sukochev, Weak compactness in Lorentz spaces, Uzb. Math. J. 1 (1993), 84-93 (in Russian).
  • [9] V. I. Chilin, P. G. Dodds, A. A. Sedaev, and F. A. Sukochev, Characterisations of Kadec-Klee properties in symmetric spaces of measurable functions, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4895-4918. MR 1390973 (97d:46031), https://doi.org/10.1090/S0002-9947-96-01782-5
  • [10] 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 (98c:46129), https://doi.org/10.1007/BF02774037
  • [11] Jacques Dixmier, von Neumann algebras, North-Holland Mathematical Library, vol. 27, North-Holland Publishing Co., Amsterdam-New York, 1981. With a preface by E. C. Lance; Translated from the second French edition by F. Jellett. MR 641217 (83a:46004)
  • [12] Peter G. Dodds, Theresa K.-Y. Dodds, and Ben de Pagter, Noncommutative Banach function spaces, Math. Z. 201 (1989), no. 4, 583-597. MR 1004176 (90j:46054), https://doi.org/10.1007/BF01215160
  • [13] Peter G. Dodds, Theresa K.-Y. Dodds, and Ben de Pagter, A general Markus inequality, Miniconference on Operators in Analysis (Sydney, 1989) Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 24, Austral. Nat. Univ., Canberra, 1990, pp. 47-57. MR 1060110 (91i:46072)
  • [14] 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 (93e:46074), https://doi.org/10.1017/S0305004100070225
  • [15] 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 (94a:46093), https://doi.org/10.2307/2154295
  • [16] P. G. Dodds, T. K. Dodds, F. A. Sukochev, and O. Ye. Tikhonov, A non-commutative Yosida-Hewitt theorem and convex sets of measurable operators closed locally in measure, Positivity 9 (2005), no. 3, 457-484. MR 2188531 (2006h:46062), https://doi.org/10.1007/s11117-005-1384-0
  • [17] P. G. Dodds and D. H. Fremlin, Compact operators in Banach lattices, Israel J. Math. 34 (1979), no. 4, 287-320 (1980). MR 570888 (81g:47037), https://doi.org/10.1007/BF02760610
  • [18] P. G. Dodds and B. de Pagter, Completely positive compact operators on non-commutative symmetric spaces, Positivity 14 (2010), no. 4, 665-679. MR 2741325 (2011k:46088), https://doi.org/10.1007/s11117-010-0073-9
  • [19] 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, https://doi.org/10.1090/S0002-9947-2012-05569-3
  • [20] 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, https://doi.org/10.1016/j.indag.2013.01.009
  • [21] P. G. Dodds, B. de Pagter, and F. Sukochev, Theory of Noncommutative Integration, unpublished monograph, to appear.
  • [22] 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 (2002f:46123), https://doi.org/10.1017/S0305004101005114
  • [23] Thierry Fack and Hideki Kosaki, Generalized $ s$-numbers of $ \tau $-measurable operators, Pacific J. Math. 123 (1986), no. 2, 269-300. MR 840845 (87h:46122)
  • [24] D. H. Fremlin, Stable subspaces of $ L^{1}+L^{\infty }$, Proc. Cambridge Philos. Soc. 64 (1968), 625-643. MR 0225154 (37 #749)
  • [25] A. Grothendieck, Topological vector spaces, Translated from the French by Orlando Chaljub, Notes on Mathematics and its Applications, Gordon and Breach Science Publishers, New York-London-Paris, 1973. MR 0372565 (51 #8772)
  • [26] U. Haagerup, H. P. Rosenthal, and F. A. Sukochev, Banach embedding properties of non-commutative $ L^p$-spaces, Mem. Amer. Math. Soc. 163 (2003), no. 776, vi+68. MR 1963854 (2004f:46076), https://doi.org/10.1090/memo/0776
  • [27] N. J. Kalton and F. A. Sukochev, Symmetric norms and spaces of operators, J. Reine Angew. Math. 621 (2008), 81-121. MR 2431251 (2009i:46118), https://doi.org/10.1515/CRELLE.2008.059
  • [28] S. G. Kreĭn, Yu. Ī. Petunīn, and E. M. Semënov, Interpolation of linear operators, Translations of Mathematical Monographs, vol. 54, American Mathematical Society, Providence, R.I., 1982. Translated from the Russian by J. Szűcs. MR 649411 (84j:46103)
  • [29] A. V. Krygin, E. M. Sheremet'ev, and F. A. Sukochev, Conjugation of weak and measure convergence in noncommutative symmetric spaces, Dokl. Akad. Nauk UzSSR (1993), no. 2, 8-9 (in Russian).
  • [30] A. V. Krygin, E. M. Sheremet'ev, and F. A. Sukochev, Convergence in measure, weak convergence and structure of subspaces in symmetric spaces of measurable operators, unpublished manuscript, 1993.
  • [31] Peter Meyer-Nieberg, Banach lattices, Universitext, Springer-Verlag, Berlin, 1991. MR 1128093 (93f:46025)
  • [32] Edward Nelson, Notes on non-commutative integration, J. Functional Analysis 15 (1974), 103-116. MR 0355628 (50 #8102)
  • [33] Erwin Neuhardt, Order properties of compact maps on $ L^p$-spaces associated with von Neumann algebras, Math. Scand. 66 (1990), no. 1, 110-116. MR 1060901 (91i:46076)
  • [34] B. de Pagter, H. Witvliet, and F. A. Sukochev, Double operator integrals, J. Funct. Anal. 192 (2002), no. 1, 52-111. MR 1918492 (2003g:46075), https://doi.org/10.1006/jfan.2001.3898
  • [35] Timur Oikhberg and Eugeniu Spinu, Domination of operators in the non-commutative setting, Studia Math. 219 (2013), no. 1, 35-67. MR 3139423, https://doi.org/10.4064/sm219-1-3
  • [36] Narcisse Randrianantoanina, Sequences in non-commutative $ L^p$-spaces, J. Operator Theory 48 (2002), no. 2, 255-272. MR 1938797 (2003h:46093)
  • [37] Yves Raynaud, On ultrapowers of non commutative $ L_p$ spaces, J. Operator Theory 48 (2002), no. 1, 41-68. MR 1926043 (2003i:46069)
  • [38] Yves Raynaud and Quanhua Xu, On subspaces of non-commutative $ L_p$-spaces, J. Funct. Anal. 203 (2003), no. 1, 149-196. MR 1996870 (2004h:46076), https://doi.org/10.1016/S0022-1236(03)00045-4
  • [39] Kazuyuki Saitô, On the preduals of $ W^{\ast } $-algebras, Tôhoku Math. J. (2) 19 (1967), 324-331. MR 0226416 (37 #2006)
  • [40] F. A. Sukochev and V. I. Chilin, A convergence criterion in regular noncommutative symmetric spaces, Izv. Akad. Nauk UzSSR Ser. Fiz.-Mat. Nauk 4 (1990), 34-39, 90 (Russian, with Uzbek summary). MR 1108770 (92e:46129)
  • [41] F. A. Sukochev and V. I. Chilin, Symmetric spaces over semifinite von Neumann algebras, Dokl. Akad. Nauk SSSR 313 (1990), no. 4, 811-815 (Russian); English transl., Soviet Math. Dokl. 42 (1991), no. 1, 97-101. MR 1080637 (92a:46075)
  • [42] Masamichi Takesaki, Theory of operator algebras. I, Springer-Verlag, New York-Heidelberg, 1979. MR 548728 (81e:46038)
  • [43] M. Terp, $ L^p$-spaces associated with von Neumann algebras, Notes, Copenhagen University (1981).
  • [44] E. V. Tokarev, Subspaces of certain symmetric spaces, Teor. Funkciĭ Funkcional. Anal. i Priložen. Vyp. 24 (1975), 156-161, iv (Russian). MR 0626854 (58 #30126)
  • [45] A. C. Zaanen, Riesz spaces. II, North-Holland Mathematical Library, vol. 30, North-Holland Publishing Co., Amsterdam, 1983. MR 704021 (86b:46001)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 46L52, 46E30, 47A30

Retrieve articles in all journals with MSC (2010): 46L52, 46E30, 47A30


Additional Information

P. G. Dodds
Affiliation: School of Computer Science, Mathematics and Engineering, Flinders University, GPO Box 2100, Adelaide 5001, Australia
Email: peter@csem.flinders.edu.au

B. de Pagter
Affiliation: Delft Institute of Applied Mathematics, Faculty EEMCS, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands
Email: b.depagter@tudelft.nl

F. Sukochev
Affiliation: School of Mathematics and Statistics, University of New South Wales, Kensington 2052, New South Wales, Australia
Email: f.sukochev@unsw.edu.au

DOI: https://doi.org/10.1090/tran/6477
Keywords: Measurable operators, uniformly absolutely continuous norm, strongly symmetric spaces
Received by editor(s): July 25, 2013
Received by editor(s) in revised form: April 29, 2014
Published electronically: September 15, 2015
Additional Notes: This work was partially supported by the Australian Research Council.
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society