A note on averaging operators. (English) Zbl 0946.46006

Jarosz, Krzysztof (ed.), Function spaces. Proceedings of the 3rd conference, Edwardsville, IL, USA, May 19-23, 1998. Providence, RI: American Mathematical Society. Contemp. Math. 232, 345-348 (1999).
If \(T\) is a positive contractive projection on the vector lattice \(E= C_0(X)\), where \(X\) is a locally compact space, then \(T\) satisfies the following identity: \(T(aTb)= T(TaTb)\) for each \(a,b\in E\). This identity was established by G. Seever in 1966. Later on C. Huijmans and B. de Pagter, and the author extended this result to semi-prime Archimedean \(f\)-algebras. In the paper under review, the author shows that the result remains valid without the assumption that \(E\) is semi-prime.
For the entire collection see [Zbl 0913.00036].


46A40 Ordered topological linear spaces, vector lattices
46E05 Lattices of continuous, differentiable or analytic functions
46B42 Banach lattices