I am currenctly working with some training exercises in operator algebra. I have to show that $\ell^1(\mathbb{Z})$ is not a $C^\ast$-algebra for the following involution defined as $(x^\ast)_n:=\overline{x_{-n}}$.
What I have in mind is that we can use the following $$x_n:=\left\{\begin{matrix} 1, & \text{ if } n=0\;\;\;\;\\ -2, & \text{ if } n=1,2\\ 0, & \text{ otherwise } \end{matrix}\right.$$ And now my goal is to show that it is not a $C^\ast$-algebra. Anyone who can help me to continue?
I assume the product on $\ell^1(\mathbb Z)$ is the pointwise product.
If it makes a $C^*$-algebra, then $\|xx^*\|=\sum_n |x_n\overline {x_{-n}}| = \|x\|^2$ for all $x$.
It's easy to find a counter example for this equality.
E.g. let $(x_n) = \chi_{\{0, 1\}}$ be the characteristic function of $\{0, 1\}$, then $\sum_n |x_n \bar x_{-n}|=1$ but $\|x\|^2= 2^2=4$.