On spaces admitting no $\ell_p$ or $c_0$ spreading model

TL;DR

The paper constructs Banach spaces that do not admit any classical spreading models like , , or , and addresses a question about quotients with separable duals, advancing understanding of Banach space structure.

Contribution

It introduces a method to produce Banach spaces with no , , or  spreading models as quotients, solving a longstanding question in the field.

Findings

Existence of spaces with no , , or  spreading models as quotients.

Construction of a separable space not admitting as a quotient any space with separable dual.

Resolution of a question by Johnson and Rosenthal regarding quotients with separable duals.

Abstract

It is shown that for each separable Banach space $X$ not admitting $\ell_1$ as a spreading model there is a space $Y$ having $X$ as a quotient and not admitting any $\ell_p$ for $1 \leq p < \infty$ or $c_0$ as a spreading model. We also include the solution to a question of W.B. Johnson and H.P. Rosenthal on the existence of a separable space not admitting as a quotient any space with separable dual.