M-ideals and split faces of the quasi state space of a non-unital ordered Banach space

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.)