Large ideals in B(L1(0,1))

Abstract: We show that the norm-closed linear span of products of two Dunford-Pettis operators on $L<sup>1(0,1)$ lies strictly between the representable and the Dunford-Pettis operators. This gives an additional proper large closed ideal in $\mathcal{B}(L<sup>1(0,1))$, answering the existence question of Johnson, Pisier and Schechtman. The quotient of the Dunford-Pettis ideal by this square ideal contains a contractively complemented isometric copy of $L<sup>1(0,1)$. We also prove that the Dunford-Pettis ideal has no right approximate identity, answering a question of Johnson and Schechtman. More precisely, a positive norm-one convolution operator $R$ satisfies $|R-RT|\geq1$ and $|R-TR|\geq1$ for every Dunford-Pettis operator $T$. The proofs combine translation averaging, Rajchman measures supported on a strongly independent set, and two atomless disintegrations of positive operators. The approximate-identity obstruction extends to every finite measure space with a nonzero atomless part. Finally, we prove that quantitative multiplication estimates for an ideal pass to all its closed power ideals.