Order ideals in order smooth $p$-normed spaces

Abstract

We generalize the notion of $M$-ideals in order smooth $\infty$-normed spaces to "smooth $p$-order ideals" in order smooth $p$-normed spaces. We show that if $V$ is an order smooth $p$-normed space and $W$ is a closed subspace of $V$, then $W$ is a smooth $p$-order ideal in $V$ if and only if $W^{\perp }$ is a smooth $p^{\prime }$-order ideal in order smooth $p^{\prime }$-normed space if and only if $W^{\perp \perp }$ is a smooth $p$-order ideal in order smooth $p$-normed space $V^{\ast \ast }$. We prove that every $L$-summand in order smooth $1$-normed space is a smooth $1$-order ideal. We find a condition under which every $M$-ideal in order smooth $\infty$-normed space is a smooth $\infty$-order ideal. We show that every $M$-ideal in order smooth $\infty$-normed space is smooth $\infty$-order ideal.