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View Those Results As Characteriza- Tions of When the Identity Mapping Is Continuous with Respect to the Unbounded/Classical Convergences Unbounded convergences in vector lattices by Mitchell A. Taylor A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Mathematics Department of Mathematical and Statistical Sciences University of Alberta © Mitchell A. Taylor, 2018 Abstract Abstract. Suppose X is a vector lattice and there is a notion σ of convergence xα x in X. Then we can speak of an “un- −→ uσ bounded” version of this convergence by saying that xα x −−→ σ if xα x u 0 for every u X+. In the literature the | − | ∧ −→ ∈ unbounded versions of the norm, order and absolute weak con- vergence have been studied. Here we create a general theory of unbounded convergence, but with a focus on uo-convergence and those convergences deriving from locally solid topologies. We also give characterizations of minimal topologies in terms of unbounded topologies and uo-convergence. At the end we touch on the theory of bibases in Banach lattices. ii iii Preface The research in this thesis is an amalgamation of the papers Un- bounded topologies and uo-convergence in locally solid vector lattices, Metrizability of minimal and unbounded topologies, Completeness of unbounded convergences, and Extending topologies to the universal σ- completion of a vector lattice. The second paper was done in collabora- tion with Marko Kandi´c, and appeared in the Journal of Mathematical Analysis and Applications. The third paper is published in the Proceed- ings of the American Mathematical Society. There are also many new results in this thesis that do not appear in the aforementioned papers. The section on bibases in Banach lattices has not yet been submitted for publication, but there are plans to do so. iv Acknowledgements I would like to thank Dr. Vladimir Troitsky, the University of Alberta and, especially, Dr. Nicole Tomczak-Jaegermann for generous financial support that relieved me from my TA duties in second year and, in turn, greatly improved the quality of my research. I would also like to thank Professor’s Bill Johnson and Thomas Schlumprecht for host- ing me for two weeks at Texas A&M University, and providing many interesting mathematical conversations. I am indebted to Professor’s Tony Lau, Vladimir Troitsky and Xinwei Yu for the role they played in shaping my mathematical perspectives, as well as helping me earn a position at the University of California, Berkeley. Most importantly, I would like to thank my grandma for making the last six years a joy. v Contents Abstract ii Preface iii Acknowledgements iv 1. The basics on uo-convergence 1 1.1. The four pillars of uo-theory 1 2. Unbounded topologies 3 3. A connection between unbounded topologies and the universal completion 9 3.1. Abriefintroductiontominimaltopologies 17 4. Miscellaneousfactsonunboundedtopologies 19 5. Products, quotients, sublattices, and completions 24 5.1. Products 24 5.2. Quotients 24 5.3. Sublattices 26 5.4. Completions 28 6. The map τ u τ 30 7→ A 6.1. Pre-Lebesguepropertyanddisjointsequences 30 6.2. σ-Lebesguetopologies 34 6.3. Entire topologies 35 6.4. Fatou topologies 36 6.5. Submetrizabilityofunboundedtopologies 41 6.6. Localconvexityandatoms 44 7. When does uAτ = uBσ? 51 8. Metrizabilityofunboundedtopologies 57 9. Locallyboundedunboundedtopologies 64 10. Athoroughstudyofminimaltopologies 66 11. Measure-theoreticresults 75 12. SomeBanachlatticeresults 80 12.1. Boundedlyuo-completeBanachlattices 81 12.2. ComparingconvergencesinBanachlattices 84 12.3. Other research that has been done in Banach lattices 89 13. σ-universal topologies 90 13.1. Extensions to the universal σ-completion 94 vi 14. Propertiesinheritedbycompletions 108 15. An alternative to uo 110 16. Producing locally solid topologies from linear topologies 116 17. Theotherendofthespectrum 120 18. Appendixonorderconvergence 125 18.1. Whyourdefinitionoforderconvergence? 125 18.2. Which properties depend on the definition of order convergence? 130 18.3. Orderconvergenceisnottopological 135 18.4. When do the order convergence definitions agree? 139 18.5. Questionsonorderconvergence 142 19. Part II 144 20. Bidecompositions 145 21. Stability of bibases 150 22. Shadesofunconditionality 152 22.1. Unconditionalbidecompositions 152 22.2. Permutabledecompositions 153 22.3. Absolutedecompositions 154 23. Examples 158 24. Embeddingintospaceswithbi-FDDs 161 25. Bibasicsequencesinsubspaces 166 26. Stronglybibasicsequences 171 27. Additionalopenquestions 175 References 178 1 1. The basics on uo-convergence We begin by recalling the fundamental results on uo-convergence that will be used freely throughout this thesis. The concept of uo- convergent nets appears intermittently in the vector lattice literature (see [Na48], [De64], [Wi77], [Ka99] as well as the notion of order∗- convergence in [Frem04] and the notion of L-convergence in [Pap64]), but it was not until the recent papers [GX14], [Gao14], [GTX17], and [GLX17] that it gained significant traction. This section will be com- pletely minimalistic, as we will just recall what will be needed for the study of unbounded topologies. There are several beautiful areas that will not be investigated, most notably, the uo-dual of [GLX17], and the applications to financial mathematics. For more on these directions we refer the reader to the work of N. Gao, D. Leung and F. Xanthos. The contributions to uo-convergence that I have made will appear in later sections. Throughout this thesis, for convenience, all vector lattices are as- sumed Archimedean. This is a minor assumption. Notice, for exam- ple, that by [AB03, Theorem 2.21] every vector lattice which admits a Hausdorff locally solid topology is automatically Archimedean. 1.1. The four pillars of uo-theory. Throughout this section, X is an (Archimedean) vector lattice. We begin with the definition of order convergence: Definition 1.1. A net (xα)α∈A in X is said to order converge to o x X, written as xα x, if there exists another net (yβ)β B in X ∈ −→ ∈ satisfying yβ 0 and for any β B there exists α A such that ↓ ∈ 0 ∈ x x y for all α α . We say that a net (x ) is order Cauchy | α − |≤ β ≥ 0 α if the double net (xα xα0 ) 0 order converges to zero. − (α,α ) This leads to the definition of uo-convergence: Definition 1.2. A net (xα) in X is said to unbounded order con- uo verge (or uo-converge) to x X, written as xα x, if xα x o ∈ −→ | − |∧ u 0 for every u X . (xα) is said to be uo-Cauchy if the double −→ ∈ + net (xα xα0 ) 0 uo-converges to zero. − (α,α ) 2 Remark 1.3. The initial motivation for uo-convergence was the fol- lowing observation: Suppose (Ω, Σ,µ) is a semi-finite measure space and X is a regular sublattice of L0(µ). Then for a sequence (xn) in X, o a.e. xn 0 in X iff (xn) is order bounded in X and xn 0. It seems −→ −−→ natural to try to remove the order boundedness condition from this uo equivalence and, indeed, uo-convergence allows one to do this: xn 0 a.e. −→ in X iff xn 0. Therefore, uo-convergence can be thought of as a −−→ generalization of convergence almost everywhere to vector lattices. For details see [GTX17]. The most important fact about uo-convergence is that it passes freely between regular sublattices. This was proved in [GTX17, Theorem 3.2], but can be traced as far back as [Pap64]: Theorem 1.4. Let Y be a sublattice of a vector lattice X. TFAE: (i) Y is regular; uo uo (ii) For any net (yα) in Y , yα 0 in Y implies yα 0 in X; −→uo −→ uo (iii) For any net (y ) in Y , y 0 in Y if and only if y 0 α α −→ α −→ in X. Theorem 1.4 is not true for order convergence. Indeed, c0 is a regular sublattice of `∞, the unit vector basis of c0 converges in order to zero in `∞, but fails to order converge in c0. The next important fact is [GTX17, Corollary 3.6]; it states that uo-convergence, although strong enough to have unique limits, is quite a weak convergence: uo Theorem 1.5. Let (xn) be a disjoint sequence in X. Then xn 0 in −→ X. Remark 1.6. See Lemma 18.10 for a very simple observation that demonstrates the extent to which uo and o-convergence crucially differ. In Corollary 15.3 we show that uo and o-convergence disagree on nets in every infinite dimensional vector lattice. The next theorem will be crucial for our analysis. One of my main tricks is to pass uo-convergence from X to the universal completion of X, prove something up there, and then pass back down. This technique 3 works very well because universally σ-complete vector lattices have remarkable properties. Here is one of them: Theorem 1.7. A sequence (xn) in a universally σ-complete vector lat- tice is uo-Cauchy iff it is order convergent. Remark 1.8. I will eventually show that uo-convergence is sequen- tially complete iff X is universally σ-complete. The next theorem is very useful, and can be found in [CL17]. It tells us exactly which subsets of X+ suffice as “test vectors” for uo- convergence: Theorem 1.9. Let X be a vector lattice and A a subset of X+. TFAE: (i) The band generated by A is X; (ii) For any x X+, x a =0 for all a A implies x =0; ∈ ∧ o ∈ (iii) For any net (xα) in X, xα a 0 for all a A implies uo | |∧ −→ ∈ x 0. α −→ All other results on uo-convergence will be pulled from their original sources. Fortunately, the literature on uo-convergence is not too scat- tered, and makes for easy reference. For standard definitions we refer to the excellent monograph [AB03]. Our notation is completely con- sistent with that book except the definition of order convergence (see the appendix), the updated terminology regarding minimal topologies, and the standard synonyms such as “order complete” and “Dedekind complete”, “vector lattice” and “Riesz space”, etc.
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