In this paper, we study some order theoretic properties of $M$-ideals in order smooth $\infty$-normed spaces. We obtain an order theoretic version of the Alfsen-Efffros' cone decomposition theorem \cite[Theorem 2.9]{AE} for order smooth $1$-normed spaces satisfying condition $(OS.1.2)$. As an application of this result, we sharpen a result on the extension of bounded positive linear functionals on subspaces of order smooth $\infty$-normed spaces. We also give two different characterizations for \emph{M-ideals} of order smooth $\infty$-normed spaces. Finally, we characterize approximate order unit spaces as those order smooth $\infty$-normed spaces $V$ that are $M$-ideals in $\tilde{V}$. Here $\tilde{V}$ is the order unit space obtained by adjoining an order unit to $V$. We obtain this result by realising a complete order smooth $\infty$-normed space $V$ as $A_0(Q(V))$, the space of continuous affine functions on $Q(V)$ vanishing at $0$. Here $Q(V)$ is the set of quasi-states of $V$. 查看全文>>