Spectral radius of a normal element in a Banach algebra

Question

I know that if $A$ is a C*-algebra, then $||x||=||x||_{sp}$ for every normal element $x\in A$. I like to find an example of a normal element $x$ in a involutive Banach algebra with $||x||\neq ||x||_{sp}$. Thanks for your regard.

Answer 1

Take $\ell_1(\mathbb{Z})$ with the $*$-operatorion given by $\delta_n^* = \delta_{-n}$. Consider the element $\tfrac{1}{2}\delta_1 + \tfrac{1}{2}\delta_2$. It has norm one but its spectral radius is 1/2.

Instead of $\mathbb{Z}$ you can take your favourite finite, non-trivial group to have a finite-dimensional example.