Abstract.
If E and F are real Banach lattices and there is an algebra and order isomorphism Φ:
(E) →(F) between their respective ordered Banach algebras of regular operators then there is a linear order isomorphism U:E→F such that Φ(T) =UTU−1 for all T ∈ (E).
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Wickstead, A. Order and algebra isomorphisms of spaces of regular operators. Math. Ann. 332, 767–774 (2005). https://doi.org/10.1007/s00208-005-0652-4
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DOI: https://doi.org/10.1007/s00208-005-0652-4