Invertibility of Hermitian Square

Question

Let $x$ be some element in a unital $C^*$-algebra. Is it true that if $x^*x$ is invertible, then $x$ is invertible?

Answer 1

No. The standard counterexample is the shift $x\colon \ell^2(\mathbb{N})\to\ell^2(\mathbb{N}), x e_n =e_{n+1}$. It satsifies $x^\ast x=1$, but $x$ is clearly not surjective.

Invertibility of $x^\ast x$ only implies the existence of a left inverse for $x$. To get invertibility of $x$ it is necessary and sufficient that both $x^\ast x$ and $xx^\ast$ are invertible.