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Let A be an Archimedean f-algebra, let $$x\in A$$ x ∈ A , and let $$\pi _{x}:A\rightarrow A$$ π x : A → A be the linear map defined by $$\pi _{x}\left( y\right) =xy,$$ π x y = x y , for all $$y\in A.$$ y ∈ A . The aim of our paper is to give necessary and sufficient conditions concerning the averaging property of (r.u) continuous projections on Archimedean f-algebras, with a range, $$R\left( T\right) ,$$ R T , a vector sublattice of A, that maps weak order units into weak order units. As an application, we prove that if A is an Archimedean f-algebra with a unit element e, T is a positive projection on A, with a range, $$R\left( T\right) ,$$ R T , a vector sublattice of A, such that T(e) is a weak order unit of A, then T is an averaging operator if and only if $$R\left( T\right) $$ R T is $$\pi _{T\left( e\right) }$$ π T e - invariant subspace. This improves considerably a result of Kuo et al. (J Math Anal Appl 303:509–521, 2005).
Bulletin of the Malaysian Mathematical Sciences Society – Springer Journals
Published: Jun 1, 2018
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