ON INTERSECTIONS OF IDEALS IN BANACH SPACES

The notion of an ideal in a Banach space was introduced by Godefroy, Kalton and Saphar in [5]. In this short note we are interested in studying finite intersections of ideals in Banach spaces. We show that for a Banach space X, if in the bidual X**, every ideal of finite codimension is the intersection of ideals of codimension one, then the same property holds in X. For a Banach space whose dual is isometric to L-1(mu) for a positive measure mu (the so called L-1-predual spaces), we show that any ideal of finite codimension is a finite intersection of ideals of codimension one. These results extend a recent result of Bandyopadhyay and Dutta [1] proved for the case of projections of norm one in spaces of continuous functions.