The order bidual of lattice ordered algebras. (English) Zbl 0549.46006

Let A be an Archimedean f-algebra with point separating order dual A’. It is shown that the space \((A')'_ n\) of all order continuous linear functionals on A’ is an Archimedean (and hence commutative!) f-algebra with respect to the Arens multiplication. Moreover, if A has a unit element, then \((A')'_ n=A''\), the whole second order dual of A. Necessary and sufficient conditions are derived for \((A')'_ n\) to be semiprime and to have a unit element respectively. It is shown that \((A')'_ n\) is semiprime if and only if the annihilator of \(\{a\in A:| a|\leq bc\) for some \(b,c\in A^+\}\) is trivial. If A is semiprime and satisfies the so-called Stone condition, then \((A')'_ {n'}\) is semiprime if and only if A has a weak approximate unit. Furthermore, \((A')'_ n\) has a unit element in this case if and only if (amongst others) every positive linear functional on A can be extended positively to its f-algebra \(Orth(A)\) of orthomorphisms. Finally, it is proved that in the latter situation Orth(A) can be embedded in \((A')'_ n\).


46A40 Ordered topological linear spaces, vector lattices
46H05 General theory of topological algebras
06F25 Ordered rings, algebras, modules
47B60 Linear operators on ordered spaces
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