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Keywords:
Freudenthal spectral theorem; band; band projection; Boolean algebra; disjointness
Freudenthal spectral theorem; band; band projection; Boolean algebra; disjointness
Summary:
Let $X$ be an Archimedean Riesz space and $\Cal P(X)$ its Boolean algebra of all band projections, and put $\Cal P_{e}=\{P e:P\in \Cal P(X)\}$ and $\Cal B_{e}=\{x\in X: x\wedge (e-x)=0\}$, $e\in X^+$. $X$ is said to have Weak Freudenthal Property (\text{$\operatorname{WFP}$}) provided that for every $e\in X^+$ the lattice $lin\, \Cal P_{e}$ is order dense in the principal band $e^{d d}$. This notion is compared with strong and weak forms of Freudenthal spectral theorem in Archimedean Riesz spaces, studied by Veksler and Lavrič, respectively. \text{$\operatorname{WFP}$} is equivalent to $X^+$-denseness of $\Cal P_{e}$ in $\Cal B_{e}$ for every $e\in X^+$, and every Riesz space with sufficiently many projections has \text{$\operatorname{WFP}$} (THEOREM).
References:
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