The Duality between Ideals of Multilinear Operators and Tensor Norms

Abstract

We develop the duality theory between ideals of multilinear operators and tensor norms that arises from the geometric approach of $\Sigma$-operators. To this end, we introduce and develop the notions of $\Sigma$-ideals of multilinear operators and $\Sigma$-tensor norms. We establish the foundations of this theory by proving a representation theorem for maximal $\Sigma$-ideals of multilinear operators by finitely generated $\Sigma$-tensor norms and a duality theorem for $\Sigma$-tensor norms. For these notions we also develop basic theory and present concrete examples.