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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.

MSC:

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