Disjointness in Partially Ordered Vector Spaces

If the vector space \({\mathcal{L}}^r(E, F)\) of all regular operators between the vector lattices E and F is ordered by the collection of its positive operators, then the Dedekind completeness of F is a sufficient condition for \({\mathcal{L}}^r(E, F)\) to be a vector lattice. \({\mathcal{L}}^r(E, F)\) and some of its subspaces might be vector lattices also in a more general situation. In the paper we deal with ordered vector spaces \({\mathcal{V}}(E, F)\) of linear operators and ask under which conditions are they vector lattices, lattice-subspaces of the ordered vector space \({\mathcal{L}}^r(E, F)\) or, in the case that \({\mathcal{L}}^r(E, F)\) is a vector lattice, sublattices or even Banach lattices when equipped with the regular norm. The answer is affirmative for many classes of operators such as compact, weakly compact, regular AM-compact, regular Dunford-Pettis operators and others if acting between appropriate Banach lattices. Then it is possible to study the finite elements in such vector lattices \({\mathcal{V}}(E, F)\), where F is not necessary Dedekind complete. In the last part of the paper there will be considered the question how the order structures of E, F and \({\mathcal{V}}(E, F)\) are mutually related. It is also shown that those rank one and finite rank operators, which are constructed by means of finite elements from E′ and F, are finite elements in \({\mathcal{V}}(E,F)\). The paper contains also some generalization of results obtained for the case \({\mathcal{V}}(E, F) = {\mathcal{L}}^r(E, F)\) in [10].

Let A" be a set, A a family of real-valued functions on X which contains the constants, pA the weak uniformity generated by A, and U{nAX) the collection of uniformly continuous functions to the real line R.The problem is how to construct U(\iAX) from A. The main result here is: For A a vector lattice, the collection of suprema of countable, finitely Aequiuniform, order-one subsets of A + is uniformly dense in U(pAX).Two less technical corollaries: If A is a vector lattice (resp., vector space), then the collection of functions which are finitely A-uniform and uniformly localfy-A (resp., uniformly locally piecewise-A) is uniformly dense in U(pAX).Further, for any A, a finitely A-uniform function is fust a composition F (a,,... ,af) for some a{.^ e A and F uniformly continuous on the range of (fli.ap) in Rf.Thus, such compositions are dense in U(iiAX).For BU(iiAX), the compositions with F e BU(Rr) are dense (B denoting bounded functions).So, in a sense, to know U(pAX) it suffices to know A and subspaces of the spaces Rp, and to know BU(pAX), A and the spaces Rr suffice.In case A is a vector lattice and A ** BA (i.e., A consists of bounded functions), the problem of describing U(pAX) in terms of A has an elegant solution: A is uniformly dense in U(pAX) (i.e., if / G U(pAX) and t > 0 there is a E A with \f(x) -a(x)\ < e for all x E A").This can be seen to be essentially equivalent to the usual Stone-Weierstrass Theorem.The result appears to have been first published in 1955 by Maak [M] and Nbeling and Bauer [NB]; we give a short proof in the course of proving our main theorem.If A contains unbounded functions, the situation is more complicated: A = U(R)\N is a "uniformly closed" vector lattice, while pA is (uniformly) discrete.With X = R, let A be the piecewise linear functions (finitely many pieces); pA is the usual uniformity on R and the closure of A consists only of continuous eventually linear functions.

A general extended functional is a functional which is allowed to take on infinite values; in other words such functionals are similar to the Lebesgue integral on the space of all integrable functions. The problem of representing a general extended functional f on a vector lattice X as an operator with finite values only has been solved in [12]. In fact, this problem has been solved for a larger class of general extended operators on an ordered vector space. The solution of this problem was given by means of an extension of the range R of the functional f to some ultrapower of R. Notice, however, that it is not always the case that a functional f can be considered as a trace of some internal functional *f:*X→R. (We remark, without going into details, that such an internal functional exists exactly in the case when the results of the nonstandard analysis can be used for investigation of the given functional.) In [12] a ‘standard’ necessary and sufficient condition was given for solving this latter problem on the existence of *f. Namely, f is a trace of an internal *f if and only if it is finitely extendable. This result makes the finite extendability problem worthy of study. The present paper is a first attempt in this direction. Simultaneously we introduce and study a more general notion of weak finite extendability that coincides with finite extendability, for instance, for vector lattices with a strong unit.

Sum and intersection of linear subspaces in a vector space over a field are fundamental operations in linear algebra. The purpose of this survey paper is to give a comprehensive approach to the sums and intersections of two linear subspaces and their orthogonal complements in the finite-dimensional complex vector space. We shall establish a variety of closed-form formulas for representing the direct sum decompositions of them-dimensional complex column vector space 𝔺mwith respect to a pair of given linear subspaces 𝒨 and 𝒩 and their operations, and use them to derive a huge amount of decomposition identities for matrix expressions composed by a pair of orthogonal projectors onto the linear subspaces. As applications, we give matrix representation for the orthogonal projectors onto the intersections of a pair of linear subspaces using various matrix decomposition identities and Moore–Penrose inverses; necessary and su˚cient conditions for two linear subspaces to be in generic position; characterization of the commutativity of a pair of orthogonal projectors; necessary and su˚cient conditions for equalities and inequalities for a pair of subspaces to hold; equalities and inequalities for norms of a pair of orthogonal projectors and their operations; as well as a collection of characterizations of EP-matrix.

Riesz homomorphisms between vector lattices are characterized by interval preserving order adjoints. Buskes, van Rooij, and van Haandel generalized the notion of a Riesz homomorphism to linear maps between general ordered vector spaces. In this paper, we discuss how far the characterization between vector lattices extends to directed ordered vector spaces and pre-Riesz spaces.

We study completions of Archimedean vector lattices relative to any nonempty set of positively homogeneous functions on finite-dimensional real vector spaces. Examples of such completions include square mean closed and geometric mean closed vector lattices, amongst others. These functional completions also lead to a universal definition of the complexification of any Archimedean vector lattice and a theory of tensor products and powers of complex vector lattices in a companion paper.

We present some of the classical inequalities in analysis in the context of Archimedean (real or complex) vector lattices and f-algebras. In particular, we prove an identity for sesquilinear maps from the Cartesian square of a vector space to a geometric mean closed Archimedean vector lattice, from which a Cauchy–Schwarz inequality follows. A reformulation of this result for sesquilinear maps with a geometric mean closed semiprime Archimedean f-algebra as codomain is also given. In addition, a sufficient and necessary condition for equality is presented. We also prove a Hölder inequality for weighted geometric mean closed Archimedean Φ-algebras, substantially improving results by K. Boulabiar and M. A. Toumi. As a consequence, a Minkowski inequality for weighted geometric mean closed Archimedean Φ-algebras is obtained.

Bartle–Dunford–Schwartz integration

Let # be a probability measure on the Bore1 field Z of the real line IR and let P = # | 1 7 4 be the product measure defined on the Borel field 2 ; ~ 1 7 6 1 7 4 1 7 4 -of the topological vector space I R ~ = I R | 1 7 4 .-. . If a = ( a , ) , e I is an element of IR ~, let P~ be the measure P translated by a i.e. P,,(A)=P(A+a) for A s Z ~. In the present paper we study the set E(P)= {aslR~~ where P ~ P , signifies that P ( A ) = 0 iff P,(A)=0. Previously E(P) had been studied by, amongst others, Dudley [4], Feldman [5], Shepp [9]. Our main result (w 3) completes theirs by showing that E(P) contains a Mazur-Orlicz vector subspace 1 ~ (for definitions see w 2) whenever it contains some 1-dimensional vector subspace and that whenever E(P) is a vector subspace then E(P)= l ~ We also give an explicit expression for the function 0. A counter-example based on one due to Dudley given in the appendix shows that E(P) need not always be a vector subspace of IR ~~ However, in w 4, we give a simple sufficient condition for E(P) to be a subspace and exhibit wide classes of examples of E(P)= l ~ for various functions 0. Our results were announced previously in [2] without proofs.

This chapter is divided into three parts. The first and most substantial of these is devoted to symplectic vector spaces. Using methods of exterior algebra which are due to Elie Cartan, we study the rank of an exterior 2-form on a real vector space as well as that of the 2-form induced on a vector subspace. By means of associated canonical bases we show that there is only one model of a symplectic form and that vector subspaces may be classified by the rank of the induced form. In mechanics one uses mainly isotropic and coisotropic subspaces, particularly Lagrangian subspaces which will be discussed here in some detail. In particular we study the reduction of a symplectic vector space.

A (partially) ordered vector space A (over ℝ) is said to be Archimedean if the negative elements a∈A− are the only ones for which {αa|α∈R+} has an upper bound. A vector subspace J of ordered vector space is said to be an order ideal if $$ a,b \in J,{\text{ }}c \in A,{\text{ }}a \leqq c \leqq b \Rightarrow c \in J. $$ (1.1)

In an Archimedean directed partially ordered vector space X, one can define the concept of a band in terms of disjointness. Bands can be studied by using a vector lattice cover Y of X. If X has an order unit, Y can be represented as a subspace of $$C(\Omega )$$ , where $$\Omega $$ is a compact Hausdorff space. We characterize bands in X, and their disjoint complements, in terms of subsets of $$\Omega $$ . We also analyze two methods to extend bands in X to $$C(\Omega )$$ and show how the carriers of a band and its extensions are related. We use the results to show that in each n-dimensional partially ordered vector space with a closed generating cone, the number of bands is bounded by $$\frac{1}{4}2^{2^n}$$ for $$n\ge 2$$ . We also construct examples of $$(n+1)$$ -dimensional partially ordered vector spaces with $$\left( {\begin{array}{c}2n n\end{array}}\right) +2$$ bands. This shows that there are n-dimensional partially ordered vector spaces that have more bands than an n-dimensional Archimedean vector lattice when $$n\ge 4$$ .

A.G. Vulih has shown how an essentially unique intrinsic multiplication can be defined in certain types of Riesz spaces (vector lattices) L. In general, the multiplication is not universally defined in L, but L can always be imbedded in a large space L# in which multiplication is universally defined. If ф is a normal integral in L, then ф can be extended to a normal integral on a large space L1(ф) in L#, and L1(ф) may be regarded as an abstract integral space. A very general form of the Radon-Nikodym theorem can be proved in L1(ф), and this can be used to give a relatively simple proof of a theorem of Segal giving a necessary and sufficient condition that the Radon-Nikodym theorem hold in a measure space. In another application, the multiplication is used to give a representation of certain Riesz spaces as rings of operators on a Hilbert space.

The Paving Conjecture in operator theory and the Feichtinger Conjecture in frame theory are both problems that are equivalent to the Kadison-Singer problem concerning extensions of pure states. In all three problems, one of the difficulties is that the natural multiplicative structure appears to be incompatible — the unique extension problem of Kadison-Singer is compatible with a linear subspace, but not a subalgebra; likewise, the pavable operators is known to be a linear subspace but not a subalgebra; the Feichtinger Conjecture does not even have a linear structure. The Paving Conjecture and the Feichtinger Conjecture both have special cases in terms of exponentials in L2[0, 1]. We introduce convolution as a multiplication to demonstrate a possible attack for these special cases.KeywordsKadison-Singer ProblemPavingLaurent operatorframeMathematics Subject Classification (2000)Primary: 46L99Secondary 46B9942B35

Chapter 17 - Metric Geometry