Bull. Malays. Math. Sci. Soc. https://doi.org/10.1007/s40840-018-0641-x Averaging Operators and Continuous Projections on f -Algebras Mohamed Ali Toumi Received: 19 January 2018 / Revised: 21 May 2018 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2018 Abstract Let A be an Archimedean f -algebra, let x ∈ A, and let π : A → A be the linear map deﬁned by π (y) = xy, for all 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 (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 (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 (T ) is 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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