Dedicated to Safak Alpay on the Occasion of His Sixtieth Birthday

Let Ir(L) denote the collection of all regular integral operators on L (1 < p <∞) with the regular norm ‖ · ‖r. Then Ir(L) is a Banach function space on X ×X with the Fatou property. In this paper we shall give two distinct descriptions of the order continuous dual, or associate space, of Ir(L). Our first description follows from a result on Ir(L). In [3] we showed that Ir(L) is equal to the Calderon space (L∞,1) 1 p′ (L∞,1) 1 p . As a consequence we derive our first description via Lozanovskii’s duality theorem. Then we will present a different description of this space. We will prove that the order ideal generated by the finite rank operators on L provided with an extension of the positive projective tensor norm is a Banach function space with the Fatou property, from which it follows that it is isometric to the order continuous dual of Ir(L).