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Completeness of unbounded convergences

Author: M. A. Taylor
Journal: Proc. Amer. Math. Soc.
MSC (2010): Primary 46A40, 46A16, 46B42
Published electronically: March 30, 2018
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Abstract: As a generalization of almost everywhere convergence to vector lattices, unbounded order convergence has garnered much attention. The concept of boundedly $ uo$-complete Banach lattices was introduced by N. Gao and F. Xanthos, and has been studied in recent papers by D. Leung, V. G. Troitsky, and the aforementioned authors. We will prove that a Banach lattice is boundedly $ uo$-complete iff it is monotonically complete. Afterwards, we study completeness-type properties of minimal topologies; minimal topologies are exactly the Hausdorff locally solid topologies in which $ uo$-convergence implies topological convergence.

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  • [AW97] Y. A. Abramovich and A. W. Wickstead, When each continuous operator is regular. II, Indag. Math. (N.S.) 8 (1997), no. 3, 281-294. MR 1622216
  • [AB80] C. D. Aliprantis and O. Burkinshaw, Minimal topologies and $ L_{p}$-spaces, Illinois J. Math. 24 (1980), no. 1, 164-172. MR 550659
  • [AB78] Charalambos D. Aliprantis and Owen Burkinshaw, Locally solid Riesz spaces, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. Pure and Applied Mathematics, Vol. 76. MR 0493242
  • [AB03] Charalambos D. Aliprantis and Owen Burkinshaw, Locally solid Riesz spaces with applications to economics, 2nd ed., Mathematical Surveys and Monographs, vol. 105, American Mathematical Society, Providence, RI, 2003. MR 2011364
  • [BL88] G. Buskes and I. Labuda, On Levi-like properties and some of their applications in Riesz space theory, Canad. Math. Bull. 31 (1988), no. 4, 477-486. MR 971576
  • [Con05] Jurie Conradie, The coarsest Hausdorff Lebesgue topology, Quaest. Math. 28 (2005), no. 3, 287-304. MR 2164373
  • [DL98] Lech Drewnowski and Iwo Labuda, Copies of $ c_0$ and $ l_\infty$ in topological Riesz spaces, Trans. Amer. Math. Soc. 350 (1998), no. 9, 3555-3570. MR 1466947
  • [Frem74] D. H. Fremlin, Topological Riesz spaces and measure theory, Cambridge University Press, London-New York, 1974. MR 0454575
  • [Gao14] Niushan Gao, Unbounded order convergence in dual spaces, J. Math. Anal. Appl. 419 (2014), no. 1, 347-354. MR 3217153
  • [GLX] N. Gao, D. Leung, and F. Xanthos, Duality for unbounded order convergence and applications, Positivity, to appear. arXiv:1705.06143 [math.FA].
  • [GTX17] N. Gao, V. G. Troitsky, and F. Xanthos, Uo-convergence and its applications to Cesàro means in Banach lattices, Israel J. Math. 220 (2017), no. 2, 649-689. MR 3666441
  • [GX14] Niushan Gao and Foivos Xanthos, Unbounded order convergence and application to martingales without probability, J. Math. Anal. Appl. 415 (2014), no. 2, 931-947. MR 3178299
  • [KT] M. Kandić and M.A. Taylor, Metrizability of minimal and unbounded topologies, preprint. arXiv:1709.05407 [math.FA].
  • [Lab84] Iwo Labuda, Completeness type properties of locally solid Riesz spaces, Studia Math. 77 (1984), no. 4, 349-372. MR 741452
  • [Lab85] Iwo Labuda, On boundedly order-complete locally solid Riesz spaces, Studia Math. 81 (1985), no. 3, 245-258. MR 808567
  • [Lab87] Iwo Labuda, Submeasures and locally solid topologies on Riesz spaces, Math. Z. 195 (1987), no. 2, 179-196. MR 892050
  • [LC18] H. Li and Z. Chen, Some Loose Ends on Unbounded Order Convergence, Positivity, to appear. arXiv:1609.09707v2 [math.FA].
  • [LC18] Hui Li and Zili Chen, Some loose ends on unbounded order convergence, Positivity 22 (2018), no. 1, 83-90. MR 3764632,
  • [MN91] Peter Meyer-Nieberg, Banach lattices, Universitext, Springer-Verlag, Berlin, 1991. MR 1128093
  • [Tay] M.A. Taylor, Unbounded topologies and uo-convergence in locally solid vector lattices, preprint. arXiv:1706.01575 [math.FA].
  • [LZ64] W. A. J. Luxemburg and A. C. Zaanen, Notes on Banach function spaces. X, XI, XII, XIII, Nederl. Akad. Wetensch. Proc. Ser. A 67=Indag. Math. 26 (1964), 493-506, 507-518, 519-529, 530-543. MR 0173168

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Additional Information

M. A. Taylor
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, T6G2G1, Canada

Keywords: $uo$-convergence, unbounded topology, minimal topology, completeness, boundedly $uo$-complete, monotonically complete, Levi.
Received by editor(s): August 28, 2017
Received by editor(s) in revised form: October 31, 2017
Published electronically: March 30, 2018
Additional Notes: The author acknowledges support from NSERC and the University of Alberta.
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2018 American Mathematical Society

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