What Remains the Same in Order Convergence Types

Chapter 1 contains a summary of results on Riesz spaces frequently used in this thesis. Chapter 2 considers the real linear space Lb(L, M) of all order bounded linear transformations from a Riesz space L into a Dedekind complete Riesz space M. The order structure of the Dedekind complete Riesz space Lb(L, M) is studied in some detail. Dual formulas for T(f+), T(f-) and T(|f|) are proved. The linear space of all extendable operators from the ideal A of L into M is denoted by Le b(A, M). Two theorems are proved: (i) If θ ≦ T is extendable, then T has a smallest positive extension Tm' given by Tm(u) = sup {T(v): v ∈ A; θ ≦ v ≦ u} for all u in L+. (ii) The mapping T →(T+)m - (T-)m from Leb(A, M) into Lb(L, M) is a Riesz isomorphism. Chapter 3 studies integral and normal integral transformations. Some of the theorems included in this chapter are: (i) If T ∈ Le b(A,M) is a normal integral, then so is Tm. (ii) If L is σ-Dedekind complete and M is super Dedekind complete, then T in Lb(L,M) is a normal integral if and only if NT = {u ∈ L: |T |(|u|) = θ} is a band of L. (iii) If L is σ-Dedekind complete and M is super Dedekind complete and if there exists a strictly positive operator for L into M, then L is super Dedekind complete. (iv) If M admits a strictly positive linear functional which is normal then the normal component Tn of the operator θ ≦ T ∈ Lb(L,M) is given by Tn(u) = inf {sup αT(uα): θ ≦ uα ↑ u} for all u in L+. Chapter 4 studies ordered topological vector spaces (E,τ) with particular emphasis on locally solid linear topological Riesz spaces. Order continuity and topological continuity are considered by introducing the properties (A,o), (A,i), (A,ii), (A,iii) and (A,iv). Some results from this chapter are: (i) If (L, τ) is a locally solid Riesz space, then (L,τ) satisfies (A,i) if every τ-closed ideal is a σ-ideal, and (L, τ) satisfies (A,ii) if every τ-closed ideal is a band. (ii) If (L,τ) is a metrizable locally solid Riesz space with (A,ii), then L satisfies the Egoroff property. (iii) If (L,τ) is a metrizable locally solid Riesz space, then both (A,i) and (A,iii) hold if (A,ii) holds. A counter example shows that this is not true for non-metrizable locally solid Riesz spaces. The fifth and final chapter considers Hausdorff locally solid Riesz spaces (L, τ). The topological completion of (L, τ) is denoted by (L^, τ^). Some results from this chapter are: (i) (L^,τ^) is a Hausdorff locally solid Riesz space with cone L^+ = L+ = the τ^-closure of L + in L^, containing L as a Riesz subspace. (ii) (L^,τ^) satisfies the (A,iii) property, if (L, τ) does. (iii) (L^,τ^) satisfies the (A,ii) property, if (L, τ) does. (iv) If τ is metrizable, then (L^,τ^) satisfies the (A,i) property if (L, τ) does. (v) If Lρ is a normed Riesz space with the (sequential) Fatou property, then L^ρ^ has the (sequential) Fatou property.

In this paper, using the concept of unbounded order convergence in Riesz spaces, we define new classes of operators, named unbounded order continuous (uo-continuous, for short) and boundedly unbounded order continuous operators. We give some conditions under which uo-continuity will be equivalent to order continuity of some operators on Riesz spaces. We show that the collection of all uo-continuous linear functionals on a Riesz space E is a band of $$E^\sim $$ .

In the present section we shall prove an important result which lies at the foundations of operator theory in Riesz spaces. Briefly stated, it says that if E and F are Riesz spaces with F Dedekind complete and T : E → F is a linear operator, then T is regular if and only if T is order bounded. In other words, using the notations introduced in section 18, we have L r (E, F) = L b (E, F). Moreover (and this is the really interesting point), the vector space L b (E, F) is now a Dedekind complete Riesz space with the set of positive operators (from E into F) as positive cone. This implies that every T ∈ L b (E,F) can be written as T = T + - T - , where T + and T - are positive operators such that T + =T∨0, T - = (-T) ∨0 and T+∧T-= 0. The case that F = ℝ is of special interest. Since ℝ is Dedekind complete, it follows that regular linear functionals on E are the same as order bounded linear functionals and the space L r (E, ℝ) = L b (E, ℝ) is a Dedekind complete Riesz space. This space, denoted by E~ for convenience, is called the order dual of E. The theorem stating that L r (E, F) = L b (E, F) is a Dedekind complete Riesz space is due to L.V. Kantorovitch (1936) in the Soviet Union and to H. Freudenthal (1936) in the Netherlands. The theorem on E~, with further results on order continuous linear functionals (see the next section), is due to F. Riesz (1937) in Hungary who, already in 1928 at the Bologna International Mathematical Congress, presented his ideas about linear functionals on certain ordered vector spaces, which is one of the reasons why lattice ordered vector spaces are now known as Riesz spaces.

Let L be a vector lattice, "(" x_α ") " be a L-valued net, and x∈L . If |x_α-x|∧u→┴o 0 for every u ∈〖 L〗_+ then it is said that the net "(" x_α ")" unbounded order converges to x and is denoted by □(x_α □(→┴uo x)) . This definition of unbounded order convergence has been extensively studied on many structures, including vector lattices, local solid vector lattices, normed lattices and lattice normed spaces. It is not possible to apply this type of convergence to general lattices due to the lack of algebraic structure. Therefore, we will use a type of convergence that is considered to be the motivation for this type of convergence, first defined as independent convergence in semi-ordered linear spaces and later called unbounded order convergence. Namely, L is a lattice, x_α is an L -valued net, and x ϵ L . If (x_α∧b )∨a order converges to (x∧b )∨a for every a,b∈L with a≤b, then it is said that "(" x_α ")" individual converges to x or unbounded order converges to x . This definition can be easily applied to general lattices. In this article, this definition will be understood as unbounded order convergence. Also, even if these two convergences are called by the same name, there is no equivalence between them for general lattices, an example of this is mentioned in this article. Let L be a partially ordered set, "(" x_α ")" be an L -valued net and x∈L (x_α) is said to be star convergent to x if every subnet of the net (x_α ) has a subnet that is order convergent to x and denoted by x_α □(→┴s x). In this paper, a new type of convergence on lattices is defined by combining unbounded order convergence (individual convergence) and star convergence. Let L be a lattice, (x_α ) a net and x∈L (x_α) is said to be unbounded star convergent to x if for every subnet (x_β) of (x_α), there exists a subnet (x_ζ) of (x_β) such that (x_ζ∧b)∨ □(a→┴o ) (x∧b)∨a for every a,b∈L with a≤b and it is denoted by x_α □(→┴us x). The differences between the new type of convergence, called unbounded star convergence, and order convergence, star convergence are demonstrated with counterexamples. The meaningfulness of the unbounded star convergence type is analyzed with these counterexamples and the implications presented. In addition, basic questions about unbounded star convergence of a given net on lattices such as convergence of a fixed net, uniqueness of the limit, convergence of the subnet of a convergent net are answered.

In page 838, line 7, we should write: For sequences in a Dedekind complete Riesz space E, if and only if there exists a sequence such that and for each (see [1, Page 17–18]).

Let M be an Abelian W*-algebra of operators on a Hilbert space H. Let M0 be the set of all linear, closed, densely defined transformations in H which commute with every unitary operator in the commutant M’ of M. A well known result of R. Pallu de Barriere states that if ɸ is a normal positive linear functional on M, then ɸ is of the form T → (Tx, x) for some x in H, where T is in M. An elementary proof of this result is given, using only those properties which are consequences of the fact that ReM is a Dedekind complete Riesz space with plenty of normal integrals. The techniques used lead to a natural construction of the class M0, and an elementary proof is given of the fact that a positive self-adjoint transformation in M0 has a unique positive square root in M0. It is then shown that when the algebraic operations are suitably defined, then M0 becomes a commutative algebra. If ReM0 denotes the set of all self-adjoint elements of M0, then it is proved that ReM0 is Dedekind complete, universally complete Riesz spaces which contains ReM as an order dense ideal. A generalization of the result of R. Pallu de la Barriere is obtained for the Riesz space ReM0 which characterizes the normal integrals on the order dense ideals of ReM0. It is then shown that ReM0 may be identified with the extended order dual of ReM, and that ReM0 is perfect in the extended sense. Some secondary questions related to the Riesz space ReM are also studied. In particular it is shown that ReM is a perfect Riesz space, and that every integral is normal under the assumption that every decomposition of the identity operator has non-measurable cardinal. The presence of atoms in ReM is examined briefly, and it is shown that ReM is finite dimensional if and only if every order bounded linear functional on ReM is a normal integral.

Let A be an ordered algebra with a unit \(\mathbf{e}\) and a cone \(A^+\). The class of order continuous elements \(A_\mathrm{n}\) of A is introduced and studied. If \(A=L(E)\), where E is a Dedekind complete Riesz space, this class coincides with the band \(L_\mathrm{n}(E)\) of all order continuous operators on E. Special subclasses of \(A_\mathrm {n}\) are considered. Firstly, the order ideal \(A_\mathbf{e}\) generated by \(\mathbf{e}\). It is shown that \(A_\mathbf{e}\) can be embedded into the algebra of continuous functions and, in particular, is a commutative subalgebra of A. If A is an ordered Banach algebra with normal cone \(A^+\) then \(A_\mathbf{e}\) is an AM-space and is closed in A. Secondly, the notion of an orthomorphism in the ordered algebra A is introduced. Among others, the conditions under which orthomorphisms are order continuous, are considered. In the second part, the main emphasis will be on the case of an ordered \(C^*\)-algebra A and, in particular, on the case of the algebra B(H), where H is an ordered Hilbert space with self-adjoint cone \(H^+\). If the cone \(A^+\) is normal then every element of \(A_\mathbf{e}\) is hermitian. In H the operations are introduced which coincide with the lattice ones when H is a Riesz space. It is shown that every regular \(T\in B(H)\) is an order continuous element and operators \(T\in (B(H))_I\) have properties which are analogous to the properties of orthomorphisms on Riesz spaces.

Let L be a σ-Dedekind complete Riesz space. In (8), H. Nakano uses an extension of the multiplication operator on a Riesz space into itself (analagous to the closed operator on a Hilbert space) to obtain a representation space for the Riesz space L. He calls such an operator a “dilatator operator on L.” More specifically, he shows that the set of all dilatator operators , when suitable operations are defined, is a Dedekind complete Riesz space which is isomorphic to the space of all functions defined and continuous on an open dense subset of some fixed totally disconnected Hausdorff space. The embedding of L in the function space is then obtained by showing that L is isomorphic to a Riesz subspace of . Moreover, when L is Dedekind complete, it is an ideal in , and the topological space is extremally disconnected.

Unbounded order convergence and application to martingales without probability

A net (xy) in a vector lattice is unbounded order convergent (uo-convergent) to 0 if u ∧ |xv| order converges to 0 for all u ≧ 0. We consider, in a Banach lattice, the relationship between weak and uo-convergence. We characterise those Banach lattices in which weak convergence implies uo-convergence and those in which uo-convergence of a bounded net implies weak convergence. Finally we combine the results to characterise those Banach lattices in which weak and uo-convergence coincide for bounded nets.

For vector lattices E and F, where F is Dedekind complete and supplied with a locally solid topology, we introduce the corresponding locally solid absolute strong operator topology on the order bounded operators {mathscr{L}}_{mathrm{ob}}(E,F) from E into F. Using this, it follows that {mathscr{L}}_{mathrm{ob}}(E,F) admits a Hausdorff uo-Lebesgue topology whenever F does. For each of order convergence, unbounded order convergence, and—when applicable—convergence in the Hausdorff uo-Lebesgue topology, there are both a uniform and a strong convergence structure on {mathscr{L}}_{mathrm{ob}}(E,F). Of the six conceivable inclusions within these three pairs, only one is generally valid. On the orthomorphisms of a Dedekind complete vector lattice, however, five are generally valid, and the sixth is valid for order bounded nets. The latter condition is redundant in the case of sequences of orthomorphisms, as a consequence of a uniform order boundedness principle for orthomorphisms that we establish. We furthermore show that, in contrast to general order bounded operators, orthomorphisms preserve not only order convergence of nets, but unbounded order convergence and—when applicable—convergence in the Hausdorff uo-Lebesgue topology as well.

A vector sublattice of the order bounded operators on a Dedekind complete vector lattice can be supplied with the convergence structures of order convergence, strong order convergence, unbounded order convergence, strong unbounded order convergence, and, when applicable, convergence with respect to a Hausdorff uo-Lebesgue topology and strong convergence with respect to such a topology. We determine the general validity of the implications between these six convergences on the order bounded operator and on the orthomorphisms. Furthermore, the continuity of left and right multiplications with respect to these convergence structures on the order bounded operators, on the order continuous operators, and on the orthomorphisms is investigated, as is their simultaneous continuity. A number of results are included on the equality of adherences of vector sublattices of the order bounded operators and of the orthomorphisms with respect to these convergence structures. These are consequences of more general results for vector sublattices of arbitrary Dedekind complete vector lattices. The special attention that is paid to vector sublattices of the orthomorphisms is motivated by explaining their relevance for representation theory on vector lattices.

In our earlier paper, Discrete-time stochastic processes on Riesz spaces, we introduced the concepts of conditional expectations, martingales and stopping times on Dedekind complete Riesz space with weak order units. Here we give a construction of stopping times from sequences in a Riesz space and are consequently able to prove a Riesz space uperossing theorem which is applicable to fuzzy processes.

In a Dedekind complete Riesz space, $E$, we show that if $(P_n)$ is a sequence of band projections in $E$ then $$\limsup\limits_{n\to \infty} P_n - \liminf\limits_{n\to \infty} P_n = \limsup\limits_{n\to \infty} P_n(I-P_{n+1}).$$ This identity is used to obtain conditional extensions in a Dedekind complete Riesz spaces with weak order unit and conditional expectation operator of the Barndorff-Nielsen and Balakrishnan-Stepanov generalizations of the First Borel-Cantelli Theorem.

Convergence of Riesz space martingales