Geometric nonlinear classification of L(μ)-spaces

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 { L p ( μ ) : p ∈ [ 1 , ∞ ] } for a localizable measure μ , which extend the known results on ℓ p -spaces. It turns out that L p ( μ ) -spaces are completely classified by their geometric structure spaces in the infinite-dimensional and localizable case, and that ℓ 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 ( μ ) -spaces with respect to the structure of Birkhoff-James orthogonality.