×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bernau, S.J, On semi-normal lattice rings, (), 613-616 · Zbl 0128.26202
[2] Beukers, F; Huijsmans, C.B; de Pagter, B, Unital embedding and complexification of ƒ-algebras, Math. Z., 183, 131-144, (1983) · Zbl 0494.06010
[3] Bigard, A; Keimel, K; Wolfenstein, S, Groupes et anneaux réticulés, () · Zbl 0384.06022
[4] Birkhoff, G, Moyennes des fonctions bornées, (), 143-153, Paris · Zbl 0041.44306
[5] Birkhoff, G, Lattice theory, (), 163-172
[6] Birkhoff, G; Pierce, R.S, Lattice ordered rings, An. acad. brasil. ciênc., 28, 41-69, (1956) · Zbl 0070.26602
[7] Brainerd, B, On the structure of averaging operators, J. math. anal. appl., 5, 347-377, (1962) · Zbl 0116.32102
[8] Dubreil-Jacotin, M.L, Etude algébrique des transformations de Reynolds, (), 9-27 · Zbl 0088.32703
[9] Huijsmans, C.B; de Pagter, B, Ideal theory in ƒ-algebras, Trans. amer. math. soc., 269, 225-245, (1982) · Zbl 0483.06009
[10] Huijsmans, C.B; de Pagter, B, Subalgebras and Riesz subspaces of an ƒ-algebra, (), 161-174 · Zbl 0534.46010
[11] Huijsmans, C.B; de Pagter, B, The order bidual of lattice ordered algebras, J. funct. anal., 59, 41-64, (1984) · Zbl 0549.46006
[12] De Fériet, J.Kampé, Sur un problème d’algèbre abstraite posé par la définition de la moyenne dans la théorie de la turbulence, Ann. soc. sci. bruxelles, 63, 156-172, (1949) · Zbl 0035.15301
[13] Kelley, J.L, Averaging operators on Cx(X), Illinois J. math., 2, 214-223, (1958) · Zbl 0080.32001
[14] Lipecki, Z; Thomsen, W, Extension of positive operators and extreme points IV, (), 269-273 · Zbl 0432.47021
[15] Lloyd, S.P, A mixing condition for extreme left invariant means, Trans. amer. math. soc., 125, 461-481, (1966) · Zbl 0164.44702
[16] Luxemburg, W.A.J; Zaanen, A.C, Riesz spaces I., (1971), North-Holland Amsterdam · Zbl 0231.46014
[17] Meyer, M, La stabilisateur d’un espace vectoriel réticulé, C. R. acad. sci. Paris, 283, 249-250, (1976) · Zbl 0334.46010
[18] Meyer, M, Quelques propriétés des homomorphismes d’espaces vectoriels réticulés, ()
[19] Moy, Shu-Teh Chen, Characterizations of conditional expectation as a transformation on function spaces, Pacific J. math., 4, 47-63, (1954) · Zbl 0055.12503
[20] de Pagter, B, A note on disjointness-preserving operators, (), 543-549 · Zbl 0541.47032
[21] Rao, M.M, Opérateurs de moyennes et moyennes conditionelles, C. R. acad. sci. Paris, 268, 795-797, (1969) · Zbl 0174.17703
[22] Reynolds, O, On the dynamical theory of incompressible viscous fluids and the determination of the criterion, Philos. trans. roy. soc, 186, 123-164, (1895) · JFM 26.0872.02
[23] Rota, G.-C, On the representation of averaging operators, Rend. math. univ. Padova, 30, 52-64, (1960) · Zbl 0096.09101
[24] Rota, G.-C, Reynolds operators, (), 70-83
[25] Schaefer, H.H, Banach lattices and positive operators, () · Zbl 0291.46008
[26] Seever, G.L, Nonnegative projections on C0(X), Pacific J. math., 17, 159-166, (1966) · Zbl 0137.10002
[27] Zaanen, A.C, Examples of orthomorphisms, J. approx. theory, 13, 192-204, (1975) · Zbl 0293.47010
[28] Zaanen, A.C, Riesz spaces II, (1983), North-Holland Amsterdam · Zbl 0519.46001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.