c © TÜB_ITAK Remarks on the Paper \on the Commutant of the Ideal Centre"

We continue with the work started in [4] and give a new sucient condition on Riesz spaces having topologically full centres for Z(E)C = Orth(E) to hold. If E is a Riesz space E, the order dual of E will be the Riesz space of all order bounded linear functionals on E. Riesz spaces considered in this note are assumed to have separating order duals. Z(E) will denote the ideal centre, Orth (E) , will denote the orthomorphisms of E: If E is a topological Riesz space E0 will denote continuous dual of E: When T: E! F is an order bounded operator between two Riesz spaces, the adjoint of T carries F into E and it will be denoted by T. In all undened terminology concerning Riesz spaces we will adhere to the denitions in [1], [5] and [8]. When the order dual E separates the points of the Riesz space E, an order bounded operator T: E! E is an orthomorphism if and only if its adjoint T : E ! E is an orthomorphism. Moreover, the operator : Orth(E) ! Orth(E); (T) = T is a one to one Riesz homomorphism [1]. The image under of the centre Z(E) will be denoted by Z(E):Z(E) is a Riesz subspace of Z(E)