A tensor norm preserving unconditionality for $\mathcal{L}_p$-spaces

Abstract

We show that, for each n ∈ N, there is an n-tensor norm a (in the sense of Grothendieck) with the surprising property that the α-tensor product ⊗ α (Y 1 ,... Y n ) has local unconditional structure for each choice of n arbitrary Cp j -spaces Yj. In fact, a is the tensor norm associated to the ideal of multiple 1-summing n-linear forms on Banach spaces.