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First page of On complemented subspaces of $L_1[0,1]$

On complemented subspaces of $L_1[0,1]$

Antonio Acuaviva

math.FA Sep 15, 2026 · v1
formalization analysis
Read PDF arXiv abstract
TL;DR
A Lean 4 formalisation accompanies the paper, verifying the main results on complemented subspaces of L_1[0,1].
Abstract
We construct two complemented subspaces of $L_1[0,1]$. The first has the Schur property but fails the Radon–Nikodým property. The second contains a copy of $\ell_2$ but no copy of $L_1[0,1]$. Neither space is isomorphic to a Banach lattice. This gives a negative answer to the complemented-subspace question of Lindenstrauss and Rosenthal. A Lean 4 formalisation of the main results accompanies the paper.
Problem

The Lindenstrauss–Rosenthal question asks whether every infinite-dimensional complemented subspace of L_1[0,1] is isomorphic to either ℓ_1 or L_1[0,1]. Understanding and classifying complemented subspaces of classical Banach spaces remains open.

Approach

Two complemented subspaces of L_1[0,1] are constructed using distributions of independent biased signs on the compact group G={-1,1}^ℕ, governed by a parameter sequence λ. Convolution operators from a semigroup and their resolvents yield projections; choosing λ_j=√log(j+1) gives a Schur example and λ_j=1 gives a non-Schur example. An interpolation-space interpretation via the semigroup generator is also given. A Lean 4 formalisation accompanies the main results.

Results

The first subspace has the Schur property but fails the Radon–Nikodým property; the second contains a copy of ℓ_2 but no copy of L_1[0,1]. Neither is isomorphic to a Banach lattice, giving a negative answer to the Lindenstrauss–Rosenthal question.