Characterizations of Riesz spaces with b-property

Abstract

A Riesz space E is said to have b-property if each subset which is order bounded in E ∼∼ is order bounded in E. The relationship between b-property and completeness, being a retract and the absolute weak topology |σ| (E ∼ , E) is studied. Perfect Riesz spaces are characterized in terms of b-property. It is shown that b-property coincides with the Levi property in Dedekind complete Frechet lattices. Mathematics Subject Classification (2000). Primary 46A40.

On Riesz spaces with b-property and strongly order bounded operators

“…Now suppose T is b-weakly compact. Then lim n T (x n ) exists for each b-bounded increasing sequence (x n ) in E + again by Proposition 1 in [5]. To show T does not preserve c 0 , it suffices to show that (T (x n )) is convergent for each increasing sequence in B E .…”

“…Then lim n T (x n ) exists for each increasing sequence in B E by Proposition 3.4.11 in [14]. To show T is b-weakly compact, it suffices to show that lim n T (x n ) exists for each b-bounded increasing sequence (x n ) in E + by Proposition 1 in [5]. This readily follows from the fact that each b-bounded sequence is norm bounded.…”

“…Proof It suffices to show that T (u n ) → 0 for each b-bounded disjoint sequence (u n ) in E + by Proposition 1 in [5]. By Lemma 128.2 in [18], there is a sequence (Q n ) of order projections on principal bands in E such that Q n ↓ 0 and Q n u n = u n for all n. Thus,…”

“…b-weakly compact operators were defined in [3] and studied in [4][5][6][7][8][9][10][11]. The space of bweakly compact operators is denoted by W b (E, F ) whereas W (E, F ) denotes the space of weakly compact operators.…”

On the weak compactness of b-weakly compact operators

“…operators, we refer the reader to [2][3][4][5]7] . Also, for the duality property of this class of operators, we refer the reader to [8].…”