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On Riesz homomorphisms in unital \(f\)-algebras. (English) Zbl 1212.06043

Summary: The main topic of the first section of this paper is the following theorem: let \(A\) be an Archimedean \(f\)-algebra with unit element \(e\), and \(T\: A\rightarrow A\) a Riesz homomorphism such that \(T^{2}(f)=T(fT(e))\) for all \(f\in A\). Then every Riesz homomorphism extension \(\widetilde {T}\) of \(T\) from the Dedekind completion \(A^{\delta }\) of \(A\) into itself satisfies \(\widetilde {T}^{2}(f)=\widetilde {T}(fT(e))\) for all \(f\in A^{\delta }\). In the second section this result is applied in several directions. As a first application it is applied to show a result about extensions of positive projections to the Dedekind completion. A second application of the above result is a new approach to the Dedekind completion of commutative \(d\)-algebras.

MSC:

06F25 Ordered rings, algebras, modules
46A40 Ordered topological linear spaces, vector lattices
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