LAUSR

Let A be an Archimedean f-algebra, let $$x\in A$$x∈A, and let $$\pi _{x}:A\rightarrow A$$πx:A→A be the linear map defined by $$\pi _{x}\left( y\right) =xy,$$πxy=xy, 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, $$R\left( T\right) ,$$RT, 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, $$R\left( T\right) ,$$RT, 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 $$R\left( T\right) $$RT is $$\pi _{T\left( e\right) }$$πTe- invariant subspace. This improves considerably a result of Kuo et al. (J Math Anal Appl 303:509–521, 2005).