Bounded Archimedean \(\ell\)-algebras and Gelfand-Neumark-Stone duality. (Q2847369)

This is a long (and very interesting) survey on lattice-ordered algebras (briefly, \(\ell\)-algebras) in terms of category. To be a little more precise, the category \(\mathbf{ubal}\) of uniformly complete bounded Archimedean \(\ell\)-algebras is investigated within the larger category \(\mathbf{bal}\) of bounded Archimedean \(\ell\)-algebras. Recall by the way that an \(\ell\)-algebra with unit \(e\) is said to be \textit{bounded} if for every \(a\in A\) there is a natural number \(n_a\) such that \(|a|\leq n_ae\).NEWLINENEWLINE Among other facts, it is shown that \(\mathbf{ubal}\), which is dually equivalent to the category of compact Hausdorff spaces, is the smallest nontrivial reflective subcategory of \(\mathbf{bal}\). Moreover, objects in \(\mathbf{ubal}\) are epicomplete in \(\mathbf{bal}\) and vice-versa. This leads to a categorical formulation of the Stone-Weiertrass Theorem for \(\mathbf{bal}\), from which Gelfand-Neumark-Stone duality is deduced. Furthermore, a special attention is paid to the subcategory of \(\mathbf{bal}\) of Speaker \(\mathbb R\)-algebras, including an equivalence with certain (complex) involution algebras.