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Commutativity of the Arens product in lattice ordered algebras. (English) Zbl 0945.46003

A lattice ordered algebra \(A\) is considered, i.e. \(A\) is a vector lattice and an associative algebra such that \(xy\geq 0\) whenever \(x\geq 0\), \(y\geq 0\). Denote by \(A'\) the band of all order bounded linear functionals on \(A\) and by \((A')_n'\) the band of all order bounded order continuous linear functionals on \(A'\). The author shows that \((A')_n'\) is Abelian whenever \(A\) is Abelian. The proof is based on an approximation of positive elements in \((A')_n'\) by elements in the canonical image \(\widehat A\) of \(A\) in \((A')_n'\).

MSC:

46A40 Ordered topological linear spaces, vector lattices
13J25 Ordered rings
46A20 Duality theory for topological vector spaces
06F25 Ordered rings, algebras, modules
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