Intro. Norm in a Wiener algebra on the unit circle defined as $$ \left\|\sum_{n\in \mathbb{Z}}a_n e^{inz}\right\| = \sum_{n\in\mathbb{Z}}|a_n|. $$ The Wiener $1/f$ theorem states that the function $F$ is invertible in the Wiener algebra if and only if $\inf_{\theta} |F(e^{i\theta})| > 0$. I know two proofs of this theorem: one is of Newman, see this short paper, the other one is of Gelfand which uses the technique of Banach algebras and maximal ideals. I think Newman's proof can be slightly changed so that the continuity of the mapping $f\mapsto f^{-1}$ will also follow (even the local Lipschitzness holds?).
Question. Is the continuity of the mapping $f\mapsto f^{-1}$ defined on the set of invertible elements can be somehow derived from the description of maximal ideals in the Wiener algebra? Is there a general theorem in the theory of Banach algebras for that?