Representation of \(KB\) spaces as Köthe spaces (Q2891125)

A \(KB\) space is any Dedekind complete Banach lattice \(X\) that fulfils the following two conditions: \((A)\;x_n\downarrow 0\) implies \(\| x_n\|\rightarrow 0;\;(B)\;x_n\uparrow\) and \(\sup\| x_n\| <\infty\) imply the existence of an \(x\in X\) such that \(x_n\uparrow x\). It is proved that, for each \(KB\) space \(X\), there exist a measure space \((T,\mathcal T,\mu)\) and a function norm \(\rho\) on \((T,\mathcal T,\mu)\) (of absolutely continuous type and having the weak Fatou property) such that \(X\) is algebraically, isometrically and lattice isomorphic to the Köthe space \(L_\rho\).