Diagonals of injective tensor products of Banach lattices with bases

Abstract

Let E be a Banach lattice with a 1-unconditional basis {ei : i ∈ N}. Denote by ∆(⊗ n,ǫ E) (resp. ∆(⊗ n,s,ǫ E)) the main diagonal space of the n-fold full (resp. symmetric) injective Banach space tensor product, and denote by ∆(⊗ n,|ǫ| E) (resp. ∆(⊗ n,s,|ǫ| E)) the main diagonal space of the n-fold full (resp. symmetric) injective Banach lattice tensor product. We show that these four main diagonal spaces are pairwise isometrically isomorphic. We also show that the tensor diagonal {ei⊗· · ·⊗ei : i ∈ N} is a 1-unconditional basic sequence in both⊗ n,ǫ E and⊗ n,s,ǫ E. ⊗ n,s,ǫ X. In this paper, by using a result obtained in [2] we show that the tensor diagonal {e i ⊗ · · · ⊗ e i : i ∈ N} is a 1-unconditional basic sequence in⊗ n,s,ǫ X and we also show that the diagonal projections on both⊗ n,ǫ X and⊗ n,s,ǫ X are contractive.