An $\ell^1$-Isometric Subspace of $C[0,1]$ with Countable Oscillating Spectrum

TL;DR

The paper addresses whether a countable oscillating spectrum forces $c_0$-type geometry for subspaces of $C[0,1]$. It constructs a closed subspace $X\subset C[0,1]$ that is isometric to $\ell^1$ and has $\Omega(X)=M$, where $M=\{0\}\cup\{1/n: n\ge1\}$. It then proves $X$ cannot be isomorphic to a subspace of $c_0$, thereby giving a negative answer to the Enflo–Gurariy–Seoane question in the countable-spectrum case. The work situates this example within Xie’s complete spectrum characterization, showing that countable spectra can arise without $c_0$-embeddability, and raises open problems about complementation and spectrum rigidity under isomorphisms, motivating further study of the interaction between oscillation and Banach-space geometry.

Abstract

In this paper we construct a closed subspace $X\subset C[0,1]$ with countable oscillating spectrum $Ω(X)$ such that $X$ is isometric to $\ell^1$. This provides a negative answer to Question~4.3 posed by Enflo, Gurariy, and Seoane in [Trans. Amer. Math. Soc. \textbf{366} (2014)].

Figures (8)

• Figure 1: Open intervals $U_{n,m} \cap (0,1)$ centred at $t_n\in\{0,1,\tfrac{1}{2},\tfrac{1}{3},\tfrac{1}{4}\}$ with $m=4$.
• Figure 2: The half-open interval $V_1=(1-\eta_1/2,\,1]$.
• Figure 3: $V_1$ (red), the interval $J_{\sigma}$ (black) fully contained in $V_1$, and its two subintervals $J_{\sigma,-1},J_{\sigma,+1}$ (black) with gap $\delta_{m+1}$.
• Figure 4: Level $1$: two intervals $J_{(-1)}$ and $J_{(+1)}$ inside $B$.
• Figure 5: Level $2$: each level-$1$ interval contains two subintervals, yielding four intervals ordered lexicographically.

Theorems & Definitions (28)

• Definition 1
• Theorem 3.1: EnfloGurariySeoane2014, Theorem 4.2
• Theorem 3.2: Xie2023, Theorems 1.1 and 1.2
• Claim 1
• proof
• Remark 1: Choice of $\eta_m$
• Remark 2: Order and cardinality of the intervals
• Remark 3
• Claim 2
• proof