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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 {aA:|a|bc for some b,cA + } 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 ' .

46A40Ordered topological linear spaces, vector lattices
46H05General theory of topological algebras
06F25Ordered algebraic structures
47B60Operators on ordered spaces