Decomposition of Precompact Operators in Ordered Locally Convex Spaces

Abstract

The concept of a-directedness in a partially ordered Banach space X was first introduced by Asimow [1], Some studies of this property have since been made by Rogalski [7] and Wickstead [8]. In [8], it is shown that a-directedness is equivalent to the order-boundedness of compact subsets of X and is relevant to the positively generating property of the spaces of compact operators into X. In this paper, we generalize the result of Rogalski and Wickstead to the setting of metrizable locally convex spaces.From now and onward, E will denote a real locally convex space partially ordered by a wedge C (i.e. C # 0, C + C £ C and R+. C £ C; if C n -C = {0}, C is called a cone). E' is the topological dual of E and is partially ordered by the dual wedge C. H(C) denotes the space of all positively homogeneous real-valued functions on C \ each of which is >v*-continuous on every equicontinuous subset of C'\ H(C) is equipped with the topology of uniform convergence on equicontinuous subsets of C . The evaluation map is obviously continuous. °U will denote the system of all neighbourhoods of 0 in E.E is said to be quasi-directed if for each U e °ll there exists V e °ll such that V is directed by U in the sense that each finite subset of V is majorized by some element of U. E is said to be additive if each U e <?/ contains some convex V e °U such that the Minkowski functional p y of V is additive on C.If F is any locally convex space, P(F, E) will denote the space of all precompact operators from F into E and K(F, E) will denote the space of compact operators. THEOREM 1. Suppose that E is pseudo-metrizable and that C is closed. Then the following statements are equivalent.(1°) Every convergent sequence in E is majorized.(2°) E is quasi-directed and every increasing Cauchy sequence in E is majorized.(4°) Every precompact subset of E is majorized.(5°) Every compact (not necessarily Hausdorff) subset of E is majorized.(6°) P(F, E) is positively generated whenever F is an additive partially ordered locally convex space.(7°) P(F, E) is positively generated whenever F is a base semi-normed space with a closed cone. (If E is Hausdorjf, it suffices to consider normed spaces F).Proof. We first show that (1°) implies that E is quasi-directed. Suppose E is not quasi-directed and take a decreasing countable neighbourhood-base (U n ) at 0 in E.

Directedness in ordered normed spaces and operators

“…In [3], it was shown that, for Banach spaces, (i) of Theorem 2 implies a formally much stronger version of directedness property: each compact subset of the open unit ball U in E is majorized by an element in U. Again some simple examples show that the implication may not hold for non-complete spaces (e.g.…”

One-sided ideals of the non-commutative Schwartz space

“…It turns out that if null sequences in a Banach algebra factor then maximal right ideals are closed-see [ In order to prove that null sequences in S factor we cannot follow the Banach-algebra proof of Varopoulos, who proved in [24] that if a Banach algebra admits a bounded approximate identity then null sequences in this algebra factor. Instead we will rely on [1] and [9]. Our proof will be divided into several steps.…”

“…The next step is to show that S sa is quasi-directed. An ordered locally convex space is (see [9]) quasi-directed if for any neighbourhood U of zero there is another neighbourhood V of zero such that every finite subset of V is majorized by some element of U . We will now prove that the ordered self-adjoint part of the non-commutative Schwartz space is quasi-directed.…”

“…The author is very indebted to Tomasz Kania for many valuable comments and discussions and to David Blecher for pointing out the reference [9].…”