Banach spaces with small spaces of operators

TL;DR

This paper introduces a method to construct Banach spaces where all operators are close to a specific algebra, leading to examples of spaces with minimal structure and unique isomorphic properties.

Contribution

It provides a novel construction technique for Banach spaces with restricted operator sets and demonstrates several unique isomorphic and structural properties.

Findings

Constructed a prime Banach space

Created a space isomorphic to its codimension-two subspaces but not to hyperplanes

Developed a space isomorphic to its cube but not to its square

Abstract

For a certain class of algebras $\cal A$ we give a method for constructing Banach spaces $X$ such that every operator on $X$ is close to an operator in $\cal A$. This is used to produce spaces with a small amount of structure. We present several applications. Amongst them are constructions of a new prime Banach space, a space isomorphic to its subspaces of codimension two but not to its hyperplanes and a space isomorphic to its cube but not to its square.