LAUSR

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 $$\tilde{V}$$V~. Here $$\tilde{V}$$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).