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 innerouter factorization. Is there some similar factorization theorem for $W^+$?

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 $\sum(1z_n) <\infty$) and $g= (z1)^2$ 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. 

