Subsequential minimality in Gowers and Maurey spaces

TL;DR

This paper constructs a subsequentially minimal hereditarily indecomposable (HI) space within a Gowers-Maurey variant, answering a previously open question about the existence of such spaces.

Contribution

It introduces a novel method to define block sequences in Gowers-Maurey spaces, demonstrating the existence of a subsequentially minimal HI space.

Findings

Existence of a subsequentially minimal HI space.

Construction of specific block sequences in Gowers-Maurey spaces.

Solution to an open problem in the structure of Banach spaces.

Abstract

We define block sequences $(x_n)$ in every block subspace of a variant of the space of Gowers and Maurey so that the map $x_{2n-1}\mapsto x_{2n} $ extends to an isomorphism. This implies the existence of a subsequentially minimal HI space, which solves a question in \cite{FR}.