On ideals in Banach spaces

Abstract

In this paper we study the notion of an ideal, which was introduced by Godefroy, Kalton and Saphar in [7] and was called "locally one complemented" in [11], for injective and projective tensor products of Banach spaces. For a Banach space X and an ideal Y in X, we show that the injective tensor product space Y ⊗_ε Z is an ideal in X ⊗_ε Z for any Banach space Z. This as a consequence gives us a way of proving some known results about intersection properties of balls and extensions of operators on injective tensor product spaces in a unified way that does not involve any vector-valued Choquet theory. We also exhibit classes of Banach spaces in which every ideal is the range of a norm one projection.