Planes in Schatten-$3$

TL;DR

This paper proves that any two-dimensional subspace of Schatten-3 space can be isometrically embedded into L3, resolving a specific case of Hanner's inequality and suggesting broader possibilities for p ≥ 1.

Contribution

It establishes the isometric embedding for Schatten-3 spaces, solving a conjecture for p=3 and proposing a generalization for all p ≥ 1.

Findings

Confirmed isometric embedding for Schatten-3 into L3

Resolved the p=3 case of Hanner's inequality for Schatten classes

Proposed conjecture for embeddings for all p ≥ 1

Abstract

We prove that any two-dimensional real subspace of Schatten-3 can be linearly isometrically embedded into $L_{3}$. This resolves the case $p = 3$ of Hanner's inequality for Schatten classes, conjectured by Ball, Carlen and Lieb. We conjecture that similar isometric embedding is possible for any $p \geq 1$.