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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 $(A')'\sb n$ 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 $(A')'\sb n=A''$, the whole second order dual of A. Necessary and sufficient conditions are derived for $(A')'\sb n$ to be semiprime and to have a unit element respectively. It is shown that $(A')'\sb n$ is semiprime if and only if the annihilator of $\{a\in A:\vert a\vert\le bc$ for some $b,c\in A\sp+\}$ is trivial. If A is semiprime and satisfies the so-called Stone condition, then $(A')'\sb {n'}$ is semiprime if and only if A has a weak approximate unit. Furthermore, $(A')'\sb n$ 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 $(A')'\sb n$.

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