On transitivity of M(r,s)-inequalities and geometry of higher duals of Banach spaces

Abstract

In this note we study the transitivity of M(r,s)-inequalities and geometry of higher duals of Banach spaces. Our main theorem shows that if X and Y are closed subspaces of a Banach space Z such that X is an ideal satisfying the M(r,s)-inequality in Y and Y is an ideal satisfying the M(r,s)-inequality in Z, then X is an ideal satisfying the M(r/(2−r),s/(2−r))-inequality in Z. This extends the corresponding result for M-ideals. As an application we show that if a Banach space X is an ideal satisfying the M(r,s)-inequality in its bidual, then X is an ideal satisfying the M(r/(r+n(1−r)),c/(r+n(1−r)))-inequality in its dual space of order 2n for every n∈N. It follows that if X is an ideal satisfying the M(1,s)-inequality in its bidual, then X has the strong uniqueness property SU in all its duals of even order. These results generalize the corresponding results for M-ideals established by  Rao [R].