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We study the relationship between weak* Dunford–Pettis and weakly (resp. M-weakly, order weakly, almost M-weakly, and almost L-weakly) operators on Banach lattices. The following is one of the major results dealing with this matter: If E and F are Banach lattices such that F is Dedekind $\sigma $-complete, then each positive weak* Dunford–Pettis operator $T:E\rightarrow F$ is weakly compact if and only if one of the following assertions is valid: (a) the norms of $E^{\prime }$ and F are order continuous; (b) E is reflexive; and (c) F is reflexive.

In this first part of a study of ordered operator spaces, we develop the basic theory of 'ordered C*-bimodules'. A crucial role is played by 'open tripotents', a JB*-triple variant of Akemann's notion of open projection.

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Some results on various types of compactness of weak* Dunford–Pettis operators on Banach lattices

ORDERED C$^\ast$-MODULES

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Some results on various types of compactness of weak* Dunford–Pettis operators on Banach lattices

ORDERED C$^\ast$-MODULES