Cube root in $ C^{*}$-algebra.

Question

Let $A$ be a $C^*\text{-algbera}$ and $x\in A$. I'm trying to show that

a)for $0<\alpha<\frac{1}{2}$, there exists $u\in A$ with $x=u(x^*x)^{\alpha}$ and $u^*u=(x^*x)^{1-2\alpha}$.

b) there exists $y\in A$ such that $x=yy^*y$ ($\text{"cube root"}$ of x) and such $y$ is unique.

Thank you for your help.

Answer 1

For the proof of first part see $C^*$-algebras and their automorphisms group by G. K. Pedersen pages 11-12, lemmas 1.4.4-1.4.5.

For the proof of the second part use previous result with $\alpha=1/3$. Then you get $u\in A$ such that $x=u(x^*x)^{1/3}$ and $u^*u=(x^*x)^{1/3}$. Hence $x=uu^*u$. Proof of uniqueness I leave to you since this is a homework.