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

Enrique García-Sánchez, David de Hevia

Abstract

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

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

Abstract

Motivated by a question posed by Freeman, Oikhberg, Pineau and Taylor, we prove that if is a compact Hausdorff space with , where , then isometrically embeds into doing stable phase retrieval (SPR). We also show that the latter cannot be extended to the case .

Paper Structure

This paper contains 9 sections, 22 theorems, 45 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation and preliminary results
  3. Summary of the results
  4. Some observations on the Main Question
  5. A simplification of the question
  6. "Nice" embeddings that preserve SPR subspaces
  7. Freeman-Oikhberg-Pineau-Taylor's proof revisited
  8. Main results
  9. Open questions

Key Result

Theorem 1.4

Let $E$ be a subspace of a real Banach lattice $X$, and let $C>0$ be a constant. $E$ does $C$-SPR if and only if $E$ does not contain any $\frac{1}{C}$-almost disjoint pair. In particular, $E$ does SPR if and only if it does not contain almost disjoint pairs.

Figures (1)

  • Figure 1: Representation of $x^{(n)}$. The points in $[1,\omega^2]$ are ordered from left to right and from the top to the bottom.

Theorems & Definitions (48)

  • Definition 1.1: FOPT
  • Definition 1.2: FOPT
  • Definition 1.3
  • Theorem 1.4
  • Proposition 1.5
  • Theorem 1.6
  • Theorem 2.1
  • proof
  • Remark 2.2
  • Proposition 2.3
  • ...and 38 more