Pre-Markov operators

In operator theory characterizing extreme points has been systematically studied in a convex set of linear operators from an algebra to another. This paper presents some new characterizations. We define pre-Markov operators and identify when the second adjoint of a linear positive operator being an extreme point in the collection of all Markov operators between the unital second order duals of two unital f-algebras. Moreover a characterization of extreme points is given in the collection of all contractive operators between unital f-algebras. In addition, we give a condition that makes an order bounded algebra homomorphism is a lattice homomorphism.

PDF

___

[1] Aliprantis CD, Burkinshaw O. Positive Operators. Orlando, FL, USA: Academic Press Inc., 1985.
[2] Arens R. The adjoint of bilinear operation. Proceedings of American Mathematical Society 1951; 2: 839-848.
[3] Ben Amor MA, Boulabiar K, El Adeb C. Extreme contractive operators on Stone f -algebras. Indagationes Mathematicae 2014; 25: 93-103.
[4] Bernau SJ, Huijsmans CB. The order bidual of almost f-algebras and d-algebras. Transactions of the American Mathematical Society 1995; 347: 4259-4275.
[5] Boulabiar K, Jaber J. The order bidual of f-algebras revisited. Positivity 2011; 15: 271-279.
[6] Huijsmans CB. The order bidual of lattice-ordered algebras. II. Journal of Operator Theory 1989; 22: 277-290.
[7] Huijsmans CB, De Pagter B. The order bidual of lattice-ordered algebras. Journal of Functional Analysis 1984; 59: 41-64.
[8] Huijsmans CB, De Pagter B. Subalgebras and Riesz subspaces of an f-algebra. Proceedings of the London Mathematical Society 1984; 48: 161-174.
[9] Jaber J. f-algebras with σ−bounded approximate unit. Positivity 2014; 18: 161-170.
[10] Jaber J. Contractive operators on semiprime f -algebras. Indagationes Mathematicae 2017; 28: 1067-1075.
[11] Luxemburg WAJ, Zaanen AC. Riesz Spaces I. Amsterdam, Netherlands: North-Holland Publishing Co., 1971.
[12] Quinn J. Intermediate Riesz spaces. Pacific Journal of Mathematics 1975; 56: 255-263.
[13] Schaefer HH. Banach Lattices and Positive Operators. Berlin, Germany: Springer, 1974.
[14] Triki A. Algebra homomorphisms in complex almost f-algebras. Commentationes Mathematicae Universitatis Carolinae 2002; 43: 23-31.
[15] Zaanen AC. Riesz Spaces II. Amsterdam, Netherlands: North-Holland Publishing Co., 1983.