In (Chil et al. Positivity, 2014), the authors claim to give 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, Positivity 18 (3): [579][580][581][582][583] 2014). In this note we show that their example is consistent with the main result of Toumi and not a counterexample.
Keywords Center of a vector lattice · Orthomorphism
Mathematics Subject Classification Primary 06F25 · 46A40We use the book [1] by Aliprantis and Burkinshaw and we refer the reader to this monograph for terminology, notation, and facts not explained or proved below.Let A be a vector lattice, let 0 ≤ v ∈ A, the sequence {a n } n≥0 in A is called (v) relatively uniformly convergent to a ∈ A if for every real number ε > 0, there exists a natural number n ε such that |a n − a| ≤ εv for all n ≥ n ε . This will be denoted by a n → a (v). If a n → a (v) for some 0 ≤ v ∈ A, then the sequence {a n } n≥1 is called (relatively) uniformly convergent to a, which is denoted by a n → a(r.u). The notion of (r.u) relatively uniform Cauchy sequence is obviously defined . A vector lattice is called elatively uniformly complete if every relatively uniform Cauchy sequence in A has a unique limit. Relatively uniform limits are unique if A is Archimedean. The relatively uniform topology is denoted by (r.u) topology.