Almost isometric ideals in Banach spaces

TL;DR

This paper introduces and investigates almost isometric ideals in Banach spaces, showing they inherit key geometric properties and characterizing Gurariy spaces through their universal ideal property.

Contribution

It defines almost isometric ideals, explores their properties, and characterizes Gurariy spaces via their universal ideal behavior in superspaces.

Findings

Almost isometric ideals inherit diameter 2 properties.

They also inherit the Daugavet property.

Gurariy spaces are characterized by being almost isometric ideals in every superspace.

Abstract

A natural class of ideals, almost isometric ideals, of Banach spaces is defined and studied. The motivation for working with this class of subspaces is our observation that they inherit diameter 2 properties and the Daugavet property. Lindenstrauss spaces are known to be the class of Banach spaces that are ideals in every superspace; we show that being an almost isometric ideal in every superspace characterizes the class of Gurariy spaces.