Locally octahedral and locally almost square K\"othe-Bochner spaces

TL;DR

This paper extends previous results by proving that K"othe-Bochner spaces are locally octahedral or locally almost square without requiring simple functions to be dense, using measurable selection theorems.

Contribution

It removes the density assumption of simple functions in K"othe-Bochner spaces for properties like local octahedrality and local almost squareness.

Findings

The result holds without the density assumption.

Uses Kuratowski-Ryll-Nardzewski Theorem for measurable selections.

Generalizes previous conditions for K"othe-Bochner spaces.

Abstract

It has been proved in [J.-D. Hardtke, J. Math. Phys. Anal. Geom. 16, no.2, 119--137 (2020)] that a K\"othe-Bochner space $E(X)$ is locally octahedral/locally almost square if $X$ has the respective property and the simple functions are dense in $E(X)$. Here we show that the result still holds true without the density assumption. The proof makes use of the Kuratowski-Ryll-Nardzewski Theorem on measurable selections.