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Averaging operators and positive contractive projections. (English) Zbl 0604.47024
The following theorem is due to J. L. Kelley (1958): if X is a locally compact Hausdorff space and \(C_ 0(X)\) denotes the Banach algebra and f- algebra of all real continuous functions on X vanishing at infinity then a positive contractive projection \(T:C_ 0(X)\to C_ 0(X)\) is averaging, i.e., \(T(f.Tg)=Tf.Tg\) for all \(f,g\in C_ 0(X)\) if and only if its range R(T) is a subalgebra. In 1966, this result was generalized, independently by S. P. Lloyd and G. L. Seever, to the effect that a positive contractive projection T on \(C_ 0(X)\) satisfies the identity. \(T(f.Tg)=T(Tf.Tg)\) for all \(f,g\in C_ 0(X).\)
In the present paper these results are generalized to the large class of f-algebras. Let A be an Archimedean semiprime f-algebra and imbed A both as a Riesz subspace and a ring ideal in its unital f-algebra Orth(A) of all orthomorphisms on A, the unit element I being the identity mapping on A. It is assumed that A satisfies the Stone condition, i.e. \(f\wedge I\in A^+\) for all \(f\in A^+\). The positive linear mapping T on A is contractive whenever \(0\leq Tf\leq I\) for all \(f\in A\), \(0\leq f\leq I\). The following results are shown to be true:
(I) if T is a positive contractive projection on A, then \(T(f.Tg)=T(Tf.Tg)\) for all f,g\(\in A.\)
(II) If T is a positive projection on A, then T is averaging if and only if R(T) is a subalgebra of A and T is contractive.
In contrast to the original, fully analytic proofs of Lloyd-Seever and Kelley, our proofs are purely algebraic and order theoretical.

47B60 Linear operators on ordered spaces
46A40 Ordered topological linear spaces, vector lattices
06F25 Ordered rings, algebras, modules
Full Text: DOI
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