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

Abstract

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 This result is originally due to G. Pisier (unpublished), but our proof is different.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.

32 pages, latex2e