Banach lattices with topologically full centre

After some general background discussion on the notion of a topologically full centre in a Banach lattice, we study two problems in which it has featured. In 1988 Orhon showed that if the centre is topologically full then it is also a maximal abelian algebra of bounded operators and asked if the converse is true. We give a short proof of his result and a counterexample to the converse. After noting that every non scalar central operator has a hyperinvariant band, we show that any hyperinvariant subspace must be an order ideal, provided the centre is topologically full and conclude with a counterexample to this in a general vector lattice setting.