Isometries on extremely non-complex Banach spaces

TL;DR

This paper constructs and analyzes extremely non-complex Banach spaces, exploring their duals and isometry groups, revealing new structural properties and relationships with Hilbert spaces.

Contribution

It introduces a method to construct extremely non-complex Banach spaces with duals containing a given space as an L-summand and studies their isometry groups.

Findings

Constructed an extremely non-complex Banach space with a dual containing a specified space as an L-summand.

Showed that the isometry group of such spaces can be minimal, while their duals can have large isometry groups.

Provided examples illustrating the diversity of isometry group structures in extremely non-complex Banach spaces.

Abstract

Given a separable Banach space $E$, we construct an extremely non-complex Banach space (i.e. a space satisfying that $\|Id + T^2\|=1+\|T^2\|$ for every bounded linear operator $T$ on it) whose dual contains $E^*$ as an $L$-summand. We also study surjective isometries on extremely non-complex Banach spaces and construct an example of a real Banach space whose group of surjective isometries reduces to $\pm Id$, but the group of surjective isometries of its dual contains the group of isometries of a separable infinite-dimensional Hilbert space as a subgroup.