Local properties of accessible injective operator ideals

Abstract

In addition to Pisier's counterexample of a non-accessible maximal Banach ideal, we will give a large class of maximal Banach ideals which {\it{are accessible}}. The first step is implied by the observation that a "good behaviour" of trace duality, which is canonically induced by conjugate operator ideals can be extended to adjoint Banach ideals, if and only if these adjoint ideals satisfy an accessibility condition (theorem 3.1). This observation leads in a natural way to a characterization of accessible injective Banach ideals, where we also recognize the appearance of the ideal of {\it{absolutely summing operators}} (prop. 4.1). By the famous {\it{Grothendieck inequality}}, every operator from $L_1$ to a Hilbert space is absolutely summing, and therefore our search for such ideals will be directed towards Hilbert space factorization - via an operator version of Grothendieck's inequality (lemma 4.2). As a consequence,we obtain a class of injective ideals, which are "quasi-accessible", and with the help of {\it{tensor stability}}, we improve the corresponding norm inequalities, to get accessibility (theorem 4.1 and 4.2). In the last chapter of this paper we give applications, which are implied by a non-trivial link of the above mentioned considerations to normed products of operator ideals.