Let $A=\mathbb C[x] $ prove there is no norm on $A$ in which it is a C* algebra

Question

Let $A=\mathbb C[x] $ prove there is no norm on A in which it makes a C* algebra.

i think this is true because the spec(a) is infinity for any $a\in A$ ? but im not sure how to prove it.

I did try verifying the axioms for a norm but im not sure what a* in this case

Answer 1

First notice that for any $a\in A $ the $spec(a)=\{\lambda \in \mathbb{C}| \lambda 1 - a \space\text{is not invertable}\} $ and $r(a)=\sup\{|\lambda| s.t \lambda \in spec(a) \}$

Theorem: if a $*$-algebra posses a norm in which it is a C*algebra, then it possesses only one such norm.

Theorem: If a is a self adjiont element of a unital C*algebra A then $||a||=r(a)$ but $r(a) = \infty $ for everything which is clearly not finite.

Hence there is no norm on A that makes it into a C* algebra.