Construct an example of a Banach $*$-algebra $\mathcal A$ such that for all $t\in [0,1]$, there exists a character $\chi$ on $\mathcal A$ such that $\|\chi\|=t$.
Find example of a non unital Banach $*$-algebra which has a closed two sided ideals, which is not closed under scalar multiplication. (such example does not exists in a $C^*$-algebra because $C^*$-algebra has approximate identity)
Find example (if possible) of a unital Banach $*$-algebra with a closed two sided ideals, but is not self adjoint.
Example of a unital Banach $*$-algebra $\mathcal A$ with a normal element $a$ such that there does not exists unital $*$-homomorphism from $C(\sigma(a))$ to $\mathcal A$ such that identity map goes to $a$.
Example (if possible) of a $*$-homomorphism from a Banach $*$-algebra to a Banach $*$-algebra which is not continuous (Codomain can't be $C^*$-algebra, else it will become contractive infact).
These are not homework problems. Such examples don't exists for a $C^*$-algebra and I am unable to prove these in a Banach $*$-algebra.