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

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)].