Geometric nonlinear classification of Lp(μ)-spaces

Abstract

In this paper, we study nonlinear classification of Banach spaces with respect to geometric structure spaces. We mainly give classification results on the family of Banach spaces $\left\{L_p\right(\mu ):p\in [1,\infty \left]\right\}$ for a localizable measure μ, which extend the known results on ${\ell }_p$-spaces. It turns out that $L_p\left(\mu \right)$-spaces are completely classified by their geometric structure spaces in the infinite-dimensional and localizable case, and that ${\ell }_p$ and $L_p$ have mutually different geometric structure spaces. Since geometric structure spaces of Banach spaces are invariants of Birkhoff-James orthogonality preservers, the main results in this paper apply to classifying $L_p\left(\mu \right)$-spaces with respect to the structure of Birkhoff-James orthogonality.