Invertible positive elements

Question

In a $C^*$-algebra ${\cal A}$, I know that $a\in {\cal A}_+$ if and only if $a=x^*x$ for some $x\in {\cal A}$.

Question: If we know that $a$ is also invertible, can we choose $x$ to be invertible? It certainly is true for matrices.

Answer 1

@MikeF's comment is certainly the default answer to your question, but here is an argument which does not need square roots.

If $a$ is invertible and positive, then it also has these properties relative to the commutative C*-algebra generated by $a$ and $1$ (can you prove this?).

Therefore you may find $x$ inside that subalgebra, and hence $x$ and $x^*$ commute.

From the expression $a=x^*x$ you deduce that $x$ is left-invertible, and from $a=xx^*$, that $x$ is right-invertible. It then follows that $x$ is invertible.