When are so happy in a vector lattice that all band preserving linear operators turn out to be in the ideal centre? This question was raised by Wickstead (Representation and duality in Representation and duality Riesz spaces. Compos Math 35(3):225–238, 1977) (though not explicitly stated) and by the first author and Chil and Meyer (On the centre of a vector lattice. Indag Math 23:167–183, 2012). The answer depends on the vector lattice in which the operator in question acts. However, in this article we focus our attention on this question. First, we give a complete description of those vector lattices $$E$$ E with the property that every orthomorphism on $$E$$ E is a central operator. Secondly, we provide a counterexample to the main result, about Wickstead’s question, in a recent paper of Toumi [see Theorem 3, When orthomorphisms are in the ideal center, in Positivity (2014, in press). (Published online: 11 Dec 2013)].
Positivity – Springer Journals
Published: Sep 16, 2014
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