Subspaces of Banach spaces with big slices

TL;DR

This paper investigates conditions under which diameter two properties in Banach spaces are inherited by subspaces, focusing on properties like slice two and diameter two, and their duals, with specific emphasis on complemented subspaces and finite-dimensional quotients.

Contribution

It establishes new criteria for the inheritance of diameter two and dual properties in Banach spaces based on complementability and quotient structure.

Findings

Diameter two properties pass to complemented subspaces with norm one projections.

Finite-dimensional kernels ensure the passage of diameter two properties.

Dual properties like octahedral norms also transfer under certain conditions.

Abstract

We study when diameter two properties pass down to subspaces. We obtain that the slice two property (respectively diameter two property, strong diameter two property) passes down from a Banach space $X$ to a subspace $Y$ whenever $Y$ is complemented by a norm one projection with finite-dimensional kernel (respectively the quotient $X/Y$ is finite dimensional, $X/Y$ is strongly regular). Also we study the same problem for dual properties of the above ones, as having octahedral, weakly octahedral or 2-rough norm.