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Completeness of unbounded convergences


Author: M. A. Taylor
Journal: Proc. Amer. Math. Soc.
MSC (2010): Primary 46A40, 46A16, 46B42
DOI: https://doi.org/10.1090/proc/14007
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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Additional Information

M. A. Taylor
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, T6G2G1, Canada
Email: mataylor@ualberta.ca

DOI: https://doi.org/10.1090/proc/14007
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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