The spectrum of a $C^*$-algebra $A$ is the set of unitary equivalence classes of irreducible $*$-representations of $A$.
The spectrum of a unital commuative Banach algebra $B$ is the set of homomorphisms from $A$ to $\Bbb C$.
When we consider the Banach algebra, the definition of spectrum is different from the spectrum of a $C^*$-algebra. Can we define the spectrum of a Bannach algebra $B$ as the set of unitary equivalence classes of representations of $B$?