Dedekind Complete Riesz Space Research Articles

Lebesgue-type convergence theorems in Banach lattices with applications to compact operators

The main result is a Lebesgue-type convergence theorem in the setting of Banach lattices, of which the classical Lebesgue dominated convergence theorem is the prime example. It is proved that whenever E is a Banach lattice which is an ideal in a Biesz space M, M having the principal projection property and the Egoroff property, and if a sequence in E is order convergent in M, then the sequence is norm convergent in E if it is contained in a set of uniformly absolutely continuous norm. This result enables one to derive compactness criteria for bounded linear operators on Banach lattices. If T: E → F ( E, F Banach lattices) is a norm bounded linear operator with T[ E] ⊂ F α and if U denotes the unit ball of E, then T is compact if and only if ( a) T[ U] is of uniformly absolutely continuous norm; ( b) For every sequence S ⊂ T[ U] the band generated by S in F α is an ideal in a Dedekind complete Riesz space M with the Egoroff property and S has a subsequence converging in order in M to an element of M. From this theorem all the known compactness criteria for order bounded linear operators on Banach lattices can be derived. In particular the well-known compactness conditions proved by W. A. J. Luxemburg and A. C. Zaanen (Math. Annalen 149, 150–180 (1963)) for kernel operators on Banach function spaces are generalized to Banach lattices.

Lebesgue-type convergence theorems in Banach lattices with applications to compact operators

The main result is a Lebesgue-type convergence theorem in the setting of Banach lattices, of which the classical Lebesgue dominated convergence theorem is the prime example. It is proved that whenever E is a Banach lattice which is an ideal in a Riesz space M, M having the principal projection property and the Egoroff property, and if a sequence in E is order convergent in M, then the sequence is norm convergent in E if it is contained in a set of uniformly absolutely continuous norm. This result enables one to derive compactness criteria for bounded linear operators on Banach lattices. If T: E → E ( E, F Banach lattices) is a norm bounded linear operator with T[E] ⊂ F a and if U denotes the unit ball of E, then T is compact if and only if (a) T[ U] is of uniformly absolutely continuous norm; (b) For every sequence S ⊂ T[U] the band generated by S in F a is an ideal in a Dede-kind complete Riesz space M with the Egoroff property and S has a subsequence converging in order in M to an element of M. From this theorem all the known compactness criteria for order bounded linear operators on Banach lattices can be derived. In particular the well-known compactness conditions proved by W. A. J. Luxemburg and A. C. Zaanen (Math. Annalen 149, 150–180 (1963)) for kernel operators on Banach function spaces are generalized to Banach lattices.

Lattice Properties of Integral Operators

In this paper we are concerned with linear operators $K:L \to M$, where L is a Riesz subspace of measurable, finite a.e. functions and M is the class of all measurable, finite a.e. functions defined by $k(x,y)$ is a measurable kernel. It will be shown that the class $I[L,M]$ of all such integral operators is a Dedekind complete Riesz space, an ideal and a band in the space of order bounded linear maps $T:L \to M$.

Lattice properties of integral operators

In this paper we are concerned with linear operators K : L → M K:L \to M , where L is a Riesz subspace of measurable, finite a.e. functions and M is the class of all measurable, finite a.e. functions defined by k ( x , y ) k(x,y) is a measurable kernel. It will be shown that the class I [ L , M ] I[L,M] of all such integral operators is a Dedekind complete Riesz space, an ideal and a band in the space of order bounded linear maps T : L → M T:L \to M .

On Order Properties of Order Bounded Transformations

W. A. J. Luxemburg and A. C. Zaanen in [7] and W. A. J. Luxemburg in [5] have studied the order properties of the order bounded linear functionals of a given Riesz space L. In this paper we consider the vector space (L, M) of the order bounded linear transformations from a given Riesz space L into a Dedekind complete Riesz space M.

The order dual of an abelian von Neumann algebra

The usual technique for dealing with an abelian W*-algebra is to consider it, via the Gelfand theory, as the algebra of all continuous complex-valued functions on an extremally disconnected compact Hausdorff space with a separating family of normal linear functionals. An alternative approach, outlined in [2] and [10], is to develop the theory within the framework of Riesz spaces (linear vector lattices) where the order properties of the self-adjoint operators play an important and natural role. It has been known for a long time that the self-adjoint part of an abelian W*-algebra is a Dedekind complete Riesz space under the natural ordering of self-adjoint operators, but it is only relatively recently that a proof of this fact has been given that is independent of the Gelfand theory, and the interested reader may consult [2] or [10] for the details. This approach is essentially foreshadowed in [6] and provides a very satisfying introduction to the theory of commutative rings of operators. From this point of view, the spectral theorem for self-adjoint operators falls naturally into place as an easy consequence of the spectral theorem of H. Freudenthal. In this paper, the line of approach via Riesz spaces is developed further and several well known results are shown to follow as elementary consequences of the order structure of the algebra.

An Extension of the Concept of the Order Dual of a Riesz Space

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.