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Abstract

We study some geometrical properties of a new structure introduced by G. Pisier: the structure of lattice subspaces. We show first that if $X$ and $Y$ are Banach lattices such that $B_{\rm r}(X,Y)=B(X,Y)$, then $X$ is an $AL$-space or $Y$ is an $AM$-space. We introduce the notion of homogeneous lattice subspace and we show that up to regular isomorphism, the only homogeneous lattice subspace of $L^{p}({\mit\Omega},\mu)$, for $2\leq p<\infty$, is $G(I)$. We also show a version of the Dvoretzky theorem for this structure. We end this paper by giving an estimate of the regular Banach–Mazur distance between some finite-dimensional lattice subspaces.