Let $X$ be a locally compact Hausdorff space, which is not compact. $C_b(X)$ is a Banach algebra of all bounded continuous functions with the sup norm. Let $C_\infty(X)$ be the Banach algebra of continuous functions such that for every $f \in C_\infty(X)$, for every $\epsilon >0$ there exists a compact subset $K$ such that if $x \notin K, |f(x)|_\infty < \epsilon$. Let $p \notin X$ and define $\tilde{X} = X \cup p$. Topology on $\tilde{X}$ is topology on $X$ plus all complements of compact subsets of $X$. Notice that $\tilde{X}$ is compact Hausdorff. Set $\mathcal{B} = C(\tilde{X})$. $\mathcal{B}$ can be identified with the closed subalgebra of $C_b(X)$ generated by all $f \in C_\infty(X)$ and the constant function 1.
Let $\mathcal{A} = C_\infty(X)$. It is Banach algebra without unit. I need to determine $\hat{\mathcal{A}}$, a set of homomorphisms into R that are not identically zero. Using the fact that any homomorphism on $C(\tilde{X})$ is of the form $\phi(f) = f(y)$ for some $y \in \tilde{X}$, I figured any homomorphism in $\widehat{C_\infty(X)}$ is also of the form $\phi(g) = g(x)$ for some $x \in X$. Is this correct?