  We characterize $M$-ideals in order smooth $\infty$-normed spaces by
extending the notion of split faces of the state space to those of the
quasi-state space. We also 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
$V.$ To prove these results, we develop an order theoretic version of the
"Alfsen-Efffros' cone decomposition theorem" for order smooth $1$-normed
spaces. (As a quick application of this result, we sharpen a result on the
extension of bounded positive linear functionals on subspaces of order smooth
$\infty$-normed spaces.)
