We prove the norm estimates for operator-valued functions on free groups supported on the words with fixed length ($f={\sum }_{\left|w\right|=l}{a}_w\otimes \lambda \left(w\right)$). Next, we replace the translations by the free generators with a free family of operators and prove inequalities of the same type.

In addition to Pisier’s counterexample of a non-accessible maximal Banach ideal, we will give a large class of maximal Banach ideals which 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...