Let $A$ be a C*-algebra, $B\subseteq A$ an abelian C*-subalgebra, $\alpha \in Aut(A)$ and $X:=Spec(B)$. Let $x\in A$ with $xb=\alpha\left(b\right)x$ for all $b\in B$. Let $f \in C_0(X)$ the positive function corresponding to $x^*x$ via the Gelfand isomorphism $B\cong C_0(X)$ (it's not difficult to show that $x^*x\in B$) and let $V\subseteq \left\{ t\in X\mid f\left(t\right)\neq0\right\} $ be a clopen (closed and open) subset.
This is the setting of the proof of Lemma 4.4 in here. It now starts with "let $e\in B $ denote the characteristic function of $V$". This confuses me because I don't see why $1_V\in C_0(X)$. The continuity of $1_V$ is clear to me since $\partial V=\emptyset$. But why vanishes $1_V$ at infinity?
You are correct: $1_V$ may not be an element of $C_0(X)$. For instance, if $A=B=C_0(X)$ and $\alpha$ is the identity, you could take $x$ to be any nowhere vanishing function on $X$ that vanishes at infinity and then $V$ will be all of $X$.
To fix the result, you should assume that $V$ is compact as well, so that $1_V$ vanishes at infinity. As far as I can tell this does not cause any problems for the rest of the paper, since Lemma 4.4 seems to only be used to prove Corollary 4.5, where $X$ is assumed to be totally disconnected. If $X$ is totally disconnected (and has more than one point), then there exists a nontrivial clopen subset $V$ which is compact.