Remarque

According to TG, IX, p. 36, cor. $2,x$ is a topological zero divisor on the left (resp. right) if and only if there exists a sequence $(z_n)$ in A such that $\|z_n\|= 1$ and such that $xz_n$ tends to 0 (resp. that $z_nx$ tends to 0) when $n\rightarrow +\infty$.

A left zero divisor (resp. right) is a topological zero divisor on the left (resp. right). Suppose that A is nonzero and unital. A topological zero divisor on the left (resp. right) $x$ is not left invertible (resp. right). Indeed, if for example $yx= 1$ and if $xz_n$ tends to 0, then $z_n=y(xz_n)$ tends to 0 and one cannot have $\|z_n\|= 1$ for all $n$.