Some loose ends on unbounded order convergence
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The notion of almost everywhere convergence has been generalized to vector lattices as unbounded order convergence, which proves to be a very useful tool in the theory of vector and Banach lattices. In this short note, we establish some new results on unbounded order convergence that tie up some loose ends. In particular, we show that every norm bounded positive increasing net in an order continuous Banach lattice is uo-Cauchy and that every uo-Cauchy net in an order continuous Banach lattice has a uo-limit in the universal completion.
KeywordsUnbounded order convergence Almost everywhere convergence Vector and Banach lattices Universal completion
Mathematics Subject Classification46A40 46B42
The authors thank Dr. Niushan Gao for many valuable discussions and thank the reviewers for carefully reading the paper and providing many suggestions. The first author is grateful to Dr. Niushan Gao for his guidance.
- 1.Abramovich, Y., Aliprantis, C.D.: An invitation to operator theory. In: Graduate Studies in Mathematics, vol. 50. American Mathematical Society, Providence, RI (2002)Google Scholar
- 6.Deng, Y., O’Brien, M., Troitsky, V.G.: Unbounded norm convergence in Banach lattices. To appear in positivity. arXiv:1605.03538 [math.FA]
- 8.Gao, N., Troitsky, V.G., Xanthos, F.: Uo-convergence and its applications to Cesàro means in Banach lattices. To appear in Israel J. Math. arXiv:1509.07914 [math.FA]
- 12.Meyer-Nieberg, P.: Banach Lattices. Universitext, Springer, Berlin (1991)Google Scholar