Problem: Let $A$ be a $C^∗$-algebra and let $x \in A$ be such that $0$ is an isolated point of the spectrum of $x^∗x$. Show that $x$ has polar decomposition in $A$, i.e. the polar part of $x$ lies in $A$.
I have no idea how to proceed, although I suspect that you have to use continuous functional calculus on $x^* x$ at some point. Any help is welcome.
Since $\{0\}$ is isolated in $\sigma(x^\ast x)$, it is also isolated in $\sigma(|x|)$ and we may define continuous $f: \sigma(|x|) \rightarrow \mathbb{R}$ s.t. $f(t) = 1/t$ when $t \neq 0$ and $f(t) = 0$ when $t = 0$. Then $f(|x|) \in A$ and you can define $u = xf(|x|)$. You can then verify that $x = u|x|$ is the polar decomposition of $x$.