Our following question is motivated by this very interesting answer

Assume that $A$ is a $C^{*}$ algebra. Put $X=\{a\otimes b \mid a,b \in G(A)\}$ where $G(A)$ is the space of all invertible elements of $A$.

Assume that there are two continuous maps $f,g: X \to A$ such that $ x=f(x)\otimes g(x),\;\;\;\forall x \in X$.

Does this imply that $A$ is a finite dimensional commmutative algebra?

Note: After a few modification the above question can be generalized to the category of Banach spaces or other similar categories.

For example on can consider the following:

"Classification of all Banach space $V$ such that for every tensor product norm with $\parallel x\otimes y \parallel= \parallel x \parallel. \parallel y \parallel$ we have two continuous maps $f,g$ from the space of simple tensors to $V$ (or from $V \bar{\otimes} V$ to $V$) with $x=f(x) \otimes g(x)$"