Identity relating to positive elements in a $C^*$-algebra

Question

Let $A$ be a unital $C^*$-algebra and $a\in A$ with $||a||\leq 1$.

How might one go about showing that, for any continuous function $f:\mathbb{R}^+\rightarrow\mathbb{C}$, we have the identity $af(a^*a)=f(aa^*)a$?

Answer 1

For $f$ a polynomial it is true. I.e. $$ a (a^*a)^n = a(a^*a)\cdots(a^*a) = (aa^*)\cdots(aa^*)a = (aa^*)^na. $$