Why the disk algebra is not a C* algebra.

Question

I'm trying to figure out why the set of bounded analytic functions on the unit disk, A(D), is not a C* algebra. The norm is the sup norm and the involution is $f(z) \to \overline{f(\bar z)}$. I want to know why the C* algebra identity fails. I've tried as a counter example the function $f(z) = e^{iz}$ but can't get this to work out. Any help would be appreciated.

Answer 1

With $f(z)=e^{i z}$, we have, since $\overline{e^{iz}}=e^{-i\,\overline z}$, $$ \|f\|^2=\sup\{|e^{i z}|^2:\ z\in\mathbb D\}=\sup\{e^{-i\overline{z}+iz}:\ z\in\mathbb D\}=\sup\{e^{-2\text{Im}\,z}:\ z\in\mathbb D\}\\ =\sup\{e^{-2b}:\ b\in[-1,1]\}=e^2. $$ And $$ f^*f (z)=\overline {e^{i\overline z}}\,e^{iz}=e^{-iz}e^{iz}=1$$ for all $z $, so $\|f^*f\|=1. $