## 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

### Keywords:

vector lattice; $$d$$-algebra; $$f$$-algebra
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