This is my first post on StackExchange, so please let me know if I should modify my question in any way, or if this is not a reasonable question to ask here. I have tried to find other questions asking similar things and have not yet found a solution to my issue.
I am trying to apply the theory of the Gelfand transform to determine the spectrum of elements of the unitization of a certain non-unital commutative Banach algebra $B$ over $\mathbb{C}$. My problem arises as below:
Suppose $B$ is a complex commutative Banach algebra without unit. I form the unitization $B' := B \oplus \mathbb{C}$, with norm $||(a,c)||_{B'} = ||a||_B+|c|$ with product $(a,c)(b,c') = (ab+cb+c'a, cc')$. To determine the spectrum of any $(a, c) \in B'$ I aim to apply the theory of the Gelfand transform. Thus, I want to find characters on $B'$, bounded Banach algebra homomorphisms from $B'$ to the complex numbers. Characters must lie in the dual of $B' = B \oplus \mathbb{C}$, which is canonically isomorphic to $B^* \oplus \mathbb{C}$.
My issue is that $B$ itself has only the trivial character mapping everything to $0$, and since we know characters on $B'$ must lie in $B^* \oplus \mathbb{C}$ we find that any character on $B'$ is of the form $(\phi, I_\mathbb{C})$, where $\phi$ is a character on $B$ and $I_\mathbb{C}$ is the identity on $\mathbb{C}$. Thus there is only one character on $B'$, and it acts as $0$ on $B$ and as identity on $\mathbb{C}$.
However, the spectrum of an element $(a,c)\in B\oplus\mathbb{C}$ must be equal to $\{ \tilde{\phi}(a,c); \tilde{\phi} \text{ a character on } B\oplus\mathbb{C}\}$, due to the theory of the Gelfand transform on commutative unital Banach algebras. Thus I find the spectrum of an element of the form $(a, 0) \in B \oplus\mathbb{C}$ is exactly $\{0\}$.
In the case I am particularly interested in I know that this cannot be the case, because I have an explicit Banach algebra homomorphism from $B'$ to the bounded operators on some Banach space preserving the identity, and the image of $(a,c)$ for $a \neq 0$, has non-singleton spectrum. Identity-preserving Banach algebra homomorphisms can only decrease the spectrum (as if an element is invertible then the image of the inverse is the inverse of the image). Thus something must have gone wrong somewhere. Can anyone help me please figure out what has gone wrong? Must it be the case that there are indeed non-trivial character on my original non-unital Banach algebra $B$, but I just can't find them?
As some background to this problem, Tomek Kania's answer to the question here: An abelian Banach algebra without characters shows that there does exist some non-unital commutative Banach algebra $V$ which has only the trivial character. However, maybe in that case the spectrum of any element of the form $(v, 0) \in V \oplus \mathbb{C}$ is actually $\{0\}$.
Thanks for getting all the way to the end of my question!