Abstract
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).
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The first author is thankful to the Department of Atomic Energy, Government of India for providing financial support.
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Ghatak, A., Karn, A.K. M-ideals and split faces of the quasi state space of a non-unital ordered Banach space. Positivity 23, 413–429 (2019). https://doi.org/10.1007/s11117-018-0614-1
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DOI: https://doi.org/10.1007/s11117-018-0614-1