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The order bidual of lattice ordered algebras. II. (English) Zbl 0763.46004

The author generalizes the result obtained earlier by himself and B. de Pagter [Part I, J. Funct. Anal. 59, No. 1, 41-64 (1984; Zbl 0549.46006)]. He shows that if \(A\) is an \(f\)-algebra with a point separating dual \(A'\), then the order bidual \(A''\) is a Dedekind complete (hence Archimedean and commutative) \(f\)-algebra with respect to an Arens multiplication. All such \(f\)-algebras are thus Arens regular. It follows that all the left and right multiplications by a singular functional in the bidual are trivial.

MSC:

46A40 Ordered topological linear spaces, vector lattices
46H99 Topological algebras, normed rings and algebras, Banach algebras
46H05 General theory of topological algebras
06F25 Ordered rings, algebras, modules
47B60 Linear operators on ordered spaces

Citations:

Zbl 0549.46006
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