Nth roots for positive elements of C*-algebra

Question

There is a C*-algebra $A$ with a positive element $a$ which means $a=a^*$ and $\sigma_A{(a)}\in [0,+\infty)$, prove that $\forall n\in \mathbb{N}\exists b\in\{\text{positive}\}:b^n=a$

Of course this is the extension of the case in positive-definite matrices, but here there aren't eigenvectors! Is Gelfand-Naimark theorem (applied to a suitable sub-algebra) sufficient by simply considering root of a positive function? Or I can apply the taylor expansion of $\sqrt[n]{z}$ which has a large convergence disk?

Answer 1

The function $t\longmapsto t^{1/n}$ is well-defined and continuous on $[0,\infty)$.

So you can just use the continuous functional calculus (which amount to having a C$^*$-isomorphism between $C^*(a)$ and $C(\sigma(a))$).