Suppose that $A$ is a unital C*-algebra and $a\in A$ is positive, that is, $a$ is normal and $\sigma(a)\subset[0,\infty)$.
Then we can define the element $a^{1/2}\in A$ to be the unique element satisfying $(a^{1/2})^{2}=a$.
On the other hand, we can consider the continuous map $f\colon\sigma(a)\to\mathbb{C}$ defined by $f(x):=\sqrt{x}$ and apply the (continuous) functional calculus. Then we get an element $f(a)\in A$.
I don't understand why the definitions of $f(a)$ and $a^{1/2}$ coincide. Can someone explain what's going on? Any help will be greatly appreciated!
By the way, I use the definitions in Murphy's book on C*-algebras and Operator Theory.
Best Answer
Suppose that $b $ is positive and $b^2=a $. Let $g(t)=t^2$. The functional calculus respects composition: that is, since $t=f (g (t))$ on $\sigma (b) $, then $b=f (g (b))$.
Then $$ b=f (g (b))=f (b^2)=f (a). $$