Completeness of unbounded convergences

Taylor, M.

doi:10.1090/proc/14007

Completeness of unbounded convergences
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by M. A. Taylor PDF

Proc. Amer. Math. Soc. 146 (2018), 3413-3423 Request permission

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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