There are four facts which clarify things a little bit:

1. The inclusion $\ell^1 {\mathbb Z} \subset C(S^1)$ preserves the spectrum. (That is Wiener's Theorem)

2. If $f = \sum_i a_i z^{i}\in \ell^1 {\mathbb Z}$ has non-negative coefficients, then $\|f\| = \|f\|_{C(S^1)}$.

3. If $f(z)>0$ for all $z \in S^1$, then $f$ has a square-root in $\ell^1 {\mathbb Z}$ by holomorphic functional calculus.

4. $2 - z+z^{-1}$ is non-negative, but has no square-root in $\ell^1{\mathbb Z}$.

There are four facts which clarify things a little bit:

1. The inclusion $\ell^1 {\mathbb Z} \subset C(S^1)$ preserves the spectrum. (That is Wiener's Theorem)

2. If $f = \sum_i a_i z^{i}\in \ell^1 {\mathbb Z}$ has non-negative coefficients, then $\|f\| = \|f\|_{C(S^1)}$.

3. If $f(z)>0$ for all $z \in S^1$, then $f$ has a square-root in $\ell^1 {\mathbb Z}$ by holomorphic functional calculus.

4. $2 - z-z^{-1}$ is non-negative, but has no square-root in $\ell^1{\mathbb Z}$.