Consider the Banach algebra $W^+=\ell^1(\mathbb{Z}^+)$, viewed upon as the analytic functions $f$ on the unit disc $\mathbb{D}$ such that $$\|f\|=\sum\_{k\ge0}|a\_k|<\infty$$ where
$$f(z)=\sum a\_kz^k$$
is the Taylor expansion of $f$. Clearly, $W^+\subset H^\infty(\mathbb{D})$. Now, it is well known that $H^\infty$ admits inner-outer factorization. Is there some similar factorization theorem for $W^+$? 



Factorization in the Wiener algebra on the unit disc.
There is no factorization in Wiener algebra, it is easy to construct a counterexample. 


Namely, if $B$ is a Blaschke factor with zeroes $z\_n$, $z\_n\to 1$ (of course $\sum(1-|z\_n|) <\infty$) and $g= (z-1)^2$ then 
$
f= Bg$ has $C^1$ boundary values, and so is in the Wiener algebra. 


On the other hand, $B$ is an inner part of $f$ (in $H^\infty$), and it is not in $W$, because it is not even continuous at $1$. 


On the other hand, if $f(z)\ne 0$ for all $z:|z|=1$, then $f$ has only finitely many zeroes in the unit disc, so the factorization is trivial: the inner part is a finite Blaschke product. 


