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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 ${\left({A}^{\text{'}}\right)}_{n}^{\text{'}}$ 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 ${\left({A}^{\text{'}}\right)}_{n}^{\text{'}}={A}^{\text{'}\text{'}}$, the whole second order dual of A. Necessary and sufficient conditions are derived for ${\left({A}^{\text{'}}\right)}_{n}^{\text{'}}$ to be semiprime and to have a unit element respectively. It is shown that ${\left({A}^{\text{'}}\right)}_{n}^{\text{'}}$ is semiprime if and only if the annihilator of $\left\{a\in A:|a|\le bc$ for some $b,c\in {A}^{+}\right\}$ is trivial. If A is semiprime and satisfies the so-called Stone condition, then ${\left({A}^{\text{'}}\right)}_{{n}^{\text{'}}}^{\text{'}}$ is semiprime if and only if A has a weak approximate unit. Furthermore, ${\left({A}^{\text{'}}\right)}_{n}^{\text{'}}$ 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 $Orth\left(A\right)$ of orthomorphisms. Finally, it is proved that in the latter situation Orth(A) can be embedded in ${\left({A}^{\text{'}}\right)}_{n}^{\text{'}}$.

##### MSC:
 46A40 Ordered topological linear spaces, vector lattices 46H05 General theory of topological algebras 06F25 Ordered algebraic structures 47B60 Operators on ordered spaces