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

Abstract

In this paper we construct a closed subspace $X\subset C[0,1]$ with countable oscillating spectrum $\Omega (X)$ 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)].

Withdrawn by the author because the proof of the main theorem contains a serious error. In fact, the interval-tree construction produces an uncountable subset of the oscillating spectrum, so the claimed countability conclusion is not valid