Left multiplication operator in Banach Algebra

Question

Suppose $A$ is a Banach algebra and $x \in A$. Consider the left multiplication operator $M_x$: $$ M_x(y) = xy. $$ Assume that $M_x$ is invertible. Does that imply that $x$ is invertible in $A$?

Answer 1

$M_x$ is invertible implies $M_x$ is surjective implies, there exists $y$ such that $M_x(y)=xy=1$. Thus $x$ has a right inverse.

$M_y$ is an open map since $M_x$ is invertible (Let $U$ be an open subset, $M_x(M_y(U))=U$ is open thus $M_y(U)$ is open). Since $M_y(0)=0$, for every $z$, there exists a constant $c: cz\in M_y(U_0)$ where $U_0$ is a neighborhood of $0$ since $M_y$ is open, thus $M_y$ is surjective, $M_y$ is injective $(yz=0$ implies $x(yz)=z=0)$ thus $M_y$ is open and bijective thus $M_y$ is invertible, there exist $z$ such that $M_y(z)=yz=1, x(yz)=x=(xy)z=z$ thus $x=z$ and $xy=yx=1$ thus $x$ is invertible.