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Let E be a Banach lattice with a 1-unconditional basis $$\{e_i: i \in \mathbb {N}\}$$ { e i : i ∈ N } . Denote by $$\Delta (\check{\otimes }_{n,\epsilon }E)$$ Δ ( ⊗ ˇ n , ϵ E ) (resp. $$\Delta (\check{\otimes }_{n,s,\epsilon }E)$$ Δ ( ⊗ ˇ n , s , ϵ E ) ) the main diagonal space of the n-fold full (resp. symmetric) injective Banach space tensor product, and denote by $$\Delta (\check{\otimes }_{n,|\epsilon |}E)$$ Δ ( ⊗ ˇ n , | ϵ | E ) (resp. $$\Delta (\check{\otimes }_{n,s,|\epsilon |}E)$$ Δ ( ⊗ ˇ n , s , | ϵ | 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 $$\{e_i\otimes \cdots \otimes e_i: i \in \mathbb {N}\}$$ { e i ⊗ ⋯ ⊗ e i : i ∈ N } is a 1-unconditional basic sequence in both $$\check{\otimes }_{n,\epsilon }E$$ ⊗ ˇ n , ϵ E and $$\check{\otimes }_{n,s,\epsilon }E$$ ⊗ ˇ n , s , ϵ E .
Positivity – Springer Journals
Published: Oct 11, 2016
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