Let $(a_i)_{i \in \mathbb{Z}} \subset \mathbb{C}$ such that $\sum_{i=-\infty}^{\infty}|a_i| < \infty$.
Then if $ |z| \le 1$ $\Rightarrow$ $\sum_{i=0}^{\infty}|a_iz^i| < \sum_{i=0}^{\infty}|a_i| < \infty$ and similarly:
If $ |z| \ge 1$ $\Rightarrow$ $\sum_{i=1}^{\infty}|a_{-i}z^{-i}| < \sum_{i=1}^{\infty}|a_{-i}| < \infty$
Therefore: we have absolute convergence of $\sum_{i=-\infty}^{\infty}a_iz^i$ if $|z| = 1$
Now suppose that we have $\sum_{i=-\infty}^{\infty}a_iz^i = 1$ for all $|z| = 1$
My question: Can we say that by uniqueness of Laurent series that: $a_0= 1$ and $a_i = 0$ if $i\neq 0$ ?