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Keywords:
vector lattice; $d$-algebra; $f$-algebra
vector lattice; $d$-algebra; $f$-algebra
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.
References:
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