Applications of order convergence in Banach lattices

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The proposal is in the theory of Banach and vector lattices. This is an area of Functional Analysis that focuses on partial order structures in Banach spaces. The proposal consists of several parts. 1. Uo-convergence (unbounded order convergence) is a derivative of order convergence. Its importance became clear after a recent series of papers where my collaborators and I established some properties that make uo-convergence an excellent tool for connecting Banach lattices with function spaces. This has led to applications to order closed convex sets, preduals of Banach lattices, and risk measures. I am going to further explore certain properties and applications of uo-convergence. J.Grobler and C.Labuschagne have recently developed several new techniques based on universal completions of vector lattices and used them to extend certain results of stochastic analysis to a vector lattice setting. I am going to explore the relationship between these techniques, uo--convergence, uo-dualss, and uo-completeness, and apply this to measure-free stochastic theory. I would also like to connect these techniques with D.Fremlin's representation of universally complete spaces as spaces of measurable functions on Boolean algebras. 3. Bibasic sequences. Basic sequences play a major role in the theory of Banach spaces. In an ongoing joint project with M.Taylor, we have been studying bibasic sequences, which are basic sequences in Banach lattices whose basis expansions converge not only in norm but also in order. We have established many exciting properties of such sequences. We proved that most classical basic sequences in Analysis are bibasic. I propose to further study bibasic sequences, as well as uo-bibasic sequences. In particular, I want to determine whether every closed sublattice of a Banach lattice contains a bibasic or a uo-bibasic sequence, and whether every order basic sequence in a sequentially order complete Banach lattice is (Schauder) basic. 4. Free Banach lattices. Free Banach lattices FBL(A) and FBL[E] have recently been constructed by B.de Pagter, A.Wickstead, A.Aviles, et al. They also found an explicit formula for the norm of FBL[E]. I found an alternative way of constructing FBL(A) and FBL[E] in [T3]. In an ongoing project with M.Taylor, P.Tradacete et al, we have used the approach of [T3] to construct free p-convex Banach lattices; we have also found a formula for its norm. I propose to work on several open questions related to FBL[E]; among others, whether the sequence (|xk|) is basic in FBL[E] whenever (xk) is basic in E. I propose to use the theory of p-multinorms to find an explicit formula for the norm of the free Banach lattice with the upper p-estimate. I am also interested in constructing free Banach lattice algebras. 5. I am going to complete writing a book about vector and Banach lattices.