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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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