The Order Continuity of the Regular Norm on Regular Operator Spaces

Abstract

We present here some sufficient conditions for the regular norm on ${ℒ}^r(E,F)$ to be order continuous, and for (${ℒ}^r(E,F),\parallel ·{\parallel }_r$) to be a KB-space. In particular we deduce a characterization of the order continuity of the regular norm using L- and M-weak compactness of regular operators. Also we characterize when the space ${ℒ}^r(E,F)$ is an $L^p$-space and is lattice isomorphic to an $L^p$-space for . Some related results are also obtained.