A Gelfand duality for compact pospaces

Abstract

It is well known that the category of compact Hausdorff spaces is dually equivalent to the category of commutative $C^\star$-algebras. More generally, this duality can be seen as a part of a square of dualities and equivalences between compact Hausdorff spaces, $C^\star$-algebras, compact regular frames and de Vries algebras. Three of these equivalences have been extended to equivalences between compact pospaces, stably compact frames and proximity frames, the fourth part of what will be a second square being lacking. We propose the category of bounded Archimedean $\ell$-semi-algebras to complete the second square of equivalences and to extend the category of $C^\star$-algebras.