The set of analytic functions on unit circle is not a C*-algebra

Question

Let $\mathbb{D}$ be the open unit disc on the complex plane and consider the set $$A=\{f\in C({\rm cl}\, {\Bbb D})\colon f \text{ is an analytic function on } {\Bbb D}\}.$$ It is certainly closed under addition and multiplication, it is also closed when endowed with the supremum norm. How to show that $A$ is not a C*-algebra? Please help me.

Answer 1

What involution do you consider? If just complex conjugation, then $x\mapsto \overline{x}$ is not even differentiable.

If you consider $f\mapsto f^*$ where $f^*(z) = \overline{f(\overline{z})}$ then it does not satisfy the C*-identity.