Amenable countable groups specifically for complex-valued functions in $l^\infty(G)$

Question

Let $G$ be a countable group, and $l^\infty(G)$ the collection of functions $f:G \to \mathbb{R}$ such that $\sup |f(x)| < \infty$. I know that $G$ is amenable if and only if for all $\epsilon > 0$ and for all finite sets $H \subset G$ there exists a nonempty finite set $K \subset G$ such that $$ \frac{|Kx \Delta K|}{|K|} < \epsilon$$for all $x \in H$. Does this change if we consider $l^\infty(G)$ to be the collection of functions $f:G \rightarrow \mathbb{C}$ such that $\sup |f(x)| < \infty$? Does $$ \frac{|Kx \setminus K|}{|K|} < \epsilon$$imply that $G$ is amenable in this specific case?(I am a bit confused about the proof using Hahn Banach) Does applying in topological structure to $G$ do anything?