| Abstrakt: |
Abstract: 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. |
