Isometric embeddings into $C(K)$-spaces doing stable phase retrieval

TL;DR

This paper investigates conditions under which certain function spaces can be isometrically embedded into others while preserving stable phase retrieval, revealing specific cases where this is possible and where it fails.

Contribution

It establishes that $C[1, ext{omega}^ ext{alpha}]$ spaces can be isometrically embedded into $C(K)$ spaces with stable phase retrieval for certain countable ordinals, and identifies cases where this does not hold.

Findings

Embedding is possible for $K^{( extalpha)} eq othing$ with $2< extalpha< extomega$.

Embedding fails for the case $ extalpha=2$.

Provides conditions linking topological properties of $K$ to stable phase retrieval capabilities.

Abstract

Motivated by a question posed by Freeman, Oikhberg, Pineau and Taylor, we prove that if $K$ is a compact Hausdorff space with $K^{(\alpha)}\neq\varnothing$, where $2<\alpha<\omega$, then $C[1,\omega^\alpha]$ isometrically embeds into $C(K)$ doing stable phase retrieval (SPR). We also show that the latter cannot be extended to the case $\alpha=2$.