Embeddings of Banach Spaces into Banach Lattices and the Gordon-Lewis Property

Details Embeddings of Banach Spaces into Banach Lattices and the Gordon-Lewis Property Embeddings of Banach Spaces into Banach Lattices and the Gordon-Lewis Property

1998-12-30

arxiv.org/abs/math/9812160v1

Mathematics

Functional Analysis

32 pages, latex2e

Scientific paper

In this paper we first show that if $X$ is a Banach space and $\alpha$ is a left invariant crossnorm on $\ell_\infty\otimes X$, then there is a Banach lattice $L$ and an isometric embedding $J$ of $X$ into $L$, so that $I\otimes J$ becomes an isometry of $\ell_\infty\otimes_\alpha X$ onto $\ell_\infty\otimes_m J(X)$. Here $I$ denotes the identity operator on $\ell_\infty$ and $\ell_\infty\otimes_m J(X)$ the canonical lattice tensor product. This result is originally due to G. Pisier (unpublished), but our proof is different. We then use this to characterize the Gordon-Lewis property $\GL$ in terms of embeddings into Banach lattices. Also other structures related to the $\GL$ are investigated.

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