The Cesàro Operator is in the Toeplitz Algebra

Let $\mathbf T$ denote the $C^*$-algebra generated by all bounded Toeplitz operators on the Hardy space of the unit disk. Barría and Halmos [\emph{Trans. Amer. Math. Soc.} \textbf{273} (1982), no.~2, 621--630] asked whether the Cesàro operator belongs to $\mathbf T$. We answer this question affirmatively.