The ordinary Wiener lemma states that if $f(x):=\sum_{n \in \mathbb{Z}} a_n \exp(inx)$ ($\sum |a_n|<\infty$) and if $f(x)\neq 0$ everywhere, then $g:=1/f$ can also be written as$$g(x)= \sum b_n \exp(inx)$$with $\sum_n |b_n|<\infty$The proof of this uses the fact, that an element $u$ of a commutative Banach Algebra is invertible iff $h(u)\neq 0$ for each complex homomorphism.
Now, I have seen a similar result used in a paper - more specifically: substitute square- matrices in place of the coefficients $a_n$, i.e. consider functions of the form$$f(x):= \sum A_n \exp(inx)$$Assume now that each $f(x)$ is an invertible matrix. Does it then follow (and if so, why) that $f^{-1}(x)$ can be expressed as$$f^{-1}(x):= \sum B_n \exp(inx)$$for some matrices satisfying $\sum |B_n|<\infty $I don't know how to proceed here, because in this case the Banach Algebra is not commutative. Can I still adapt the proof of the original Wiener Lemma somehow?
