On the centre of a vector lattice (Q441339)

The centre \(Z(E)\) of a vector lattice \(E\) consists of all linear operators on \(E\) which are bounded in order by some multiple of the identity. A vector sublattice \(F\) of \(E\) is said to be order dense in \(E\) if, for every positive \(u\), there exists \(v\) in \(F\) such that \(0 < v \leq u\). Let \(E\) be a locally convex vector lattice and \(A\) be a vector sublattice of \(Z(E)\). Then, under some conditions, \(A\) separates some dense vector sublattice \(F\) of \(E\). Special attention is paid to the case \(E = L^p\).