This question is a refinement of my previous question.
Let $X$ be a compact metric space, and let $B$ be a bounded Banach Disk in $C(X)$ such that for every $x\in X$ there is $f\in B$ with $f(x)\ne 0$.
Let $M=\{u\in C(X),~ uf\in B,~\forall f\in B\}$ and let $N=\{u\in C(X),~ uf\in \overline{B},~\forall f\in \overline{B}\}$.
Since multiplication $(f,g)\to fg$ is a continuous operation on $C(X)$, it follows that $N$ is closed in $C(X)$ and $M\subset N$. Thus, $\overline{M}\subset N$.
Is it true that $N=\overline{M}$?
A Banach Disk in a vector space is a convex balanced set such that its Minkowski functional is a complete norm on its span.
