TITLE: Invertible positive elements
QUESTION [5 upvotes]: 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.

REPLY [5 votes]: @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.