I am looking for an element $(a_n)_{n\in\mathbb Z}\in\ell^1(\mathbb Z)$ with the property $$ \lVert(a_n)\rVert_{\ell^1(\mathbb Z)} > \lVert\sum_{n\in\mathbb Z}a_n z^n\rVert_\infty, $$ where the norm on the right hand side denotes the sup-norm on $\mathcal C(\mathbb T)$ ($\mathbb T$ is the 1-Torus).
Motivation: I want to prove that the Gelfand transformation on $\ell^1(\mathbb Z)$ is not isometric.