Diagonals of injective tensor products of Banach lattices with bases

Diagonals of injective tensor products of Banach lattices with bases 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 . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Diagonals of injective tensor products of Banach lattices with bases

, Volume 21 (3) – Oct 11, 2016
14 pages

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Publisher
Springer International Publishing
Copyright
Copyright © 2016 by Springer International Publishing
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-016-0447-8
Publisher site
See Article on Publisher Site

Abstract

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 .

Journal

PositivitySpringer Journals

Published: Oct 11, 2016

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