Order ideals in order smooth $p$-normed spaces

TL;DR

This paper extends the concept of $M$-ideals to smooth $p$-order ideals in order smooth $p$-normed spaces, establishing duality relations and characterizing when certain subspaces are such ideals.

Contribution

It introduces smooth $p$-order ideals, proves duality conditions, and characterizes $L$-summands and $M$-ideals in order smooth $p$-normed spaces.

Findings

Characterization of smooth $p$-order ideals via duality.

Every $L$-summand in order smooth 1-normed space is a smooth 1-order ideal.

Every $M$-ideal in order smooth $ty$-normed space is a smooth $ty$-order ideal.

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 $p'$-order ideal in order smooth $p'$-normed space if and only if $W^{\perp\perp}$ is a smooth $p$-order ideal in order smooth $p$-normed space $V^{**}$. 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.