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Zbl 0549.46006
Huijsmans, C.B.; de Pagter, B.
The order bidual of lattice ordered algebras.
(English)
[J] J. Funct. Anal. 59, 41-64 (1984). ISSN 0022-1236

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 $(A')'\sb n$.
MSC 2000:
*46A40 Ordered topological linear spaces
46H05 General theory of topological algebras
06F25 Ordered algebraic structures
47B60 Operators on ordered spaces

Keywords: second order dual; Archimedean f-algebra; order continuous linear functionals; Arens multiplication; semiprime; annihilator; Stone condition; weak approximate unit; orthomorphisms

Cited in: Zbl 0763.46004 Zbl 0616.46006

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