Averaging operators and positive contractive projections
type: Item
instance of: scholarly article
title: Averaging operators and positive contractive projections
zbMATH Open document ID: 0604.47024
DOI: 10.1016/0022-247X(86)90340-9
author: Charles B. Huijsmans
author: B. de Pagter
published in: Journal of Mathematical Analysis and Applications
publication date: 1986-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 47B60
Mathematics Subject Classification ID: 46A40
Mathematics Subject Classification ID: 06F25
zbMATH DE Number: 3975711
zbMATH Keywords: averaging operator
zbMATH Keywords: multiplicatively closed
zbMATH Keywords: f-algebra
zbMATH Keywords: positive contractive projection
zbMATH Keywords: Riesz subspace
zbMATH Keywords: unital f-algebra
zbMATH Keywords: Stone condition
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3975711