In Murphy, Theorem 2.1.10, they say that any abelian non-zero C*-algebra $A$ is $\ast$-isomorphic to $C_{0}(\Sigma_{A})$, where $\Sigma_{A}$ is a maximal ideal space.
In Conway, Corollary VIII.2.2, they say that an abelian C*-algebra $A$ without unit is $\ast$-isomorphic to $C_{0}(\Sigma_{A})$. (And Theorem VIII.2.1 says that a unital abelian C*-algebra $A$ is $\ast$-isomorphic to $C(\Sigma_{A})$.)
Why doesn't the literature agree with each other? In particular: why doesn't Murphy need 'non-unital' and why doesn't Conway need 'non-zero'?
Recall: $C_{0}(X)$ is the space of continuous functions that vanish at infinity.
Best Answer
If $A$ is an Abelian $C^\ast$-algebra then $A$ is $\ast$-isomorphic to $C_0(\Sigma_A)$, if $A$ is unital then $\Sigma_A$ is compact and so we also have that $C_0(\Sigma_A)=C(\Sigma_A)$. The statement in Conway's book is phrased in a weird way, since the "without unit" is superfluous, maybe he wanted to stress that in the unital case you can use $C(\Sigma_A)$ instead, even though it makes no difference.
As for the zero $C^\ast$-algebra it is a matter of convention, if $A=\{0\}$ then $\Sigma_A=\varnothing$ and $C(\Sigma_A)$ only contains the empty function so it could be considered as a zero dimensional vector space and be isomorphic to $A$, but for example some authors insist on a topological space being nonempty, so they'd exclude this, I'm not sure about the conventions used in Murphy's book.