The kernel of all invariant means

(Essentially from Narutaka's comments)

The trivial answer would be the closed linear span of $$\{f-\lambda_g(f):\;g\in G,\;f\in\ell^\infty(G).$$

For a non-amenable group this is all of $\ell^\infty(G)$. For an amenable group with Følner net $(F_i)$, this is the set of $f$ such that $$\lim_{i\to\infty}\sup_{x\in G}\frac{\left|\sum_{y\in F_ix}f(y)\right|}{|F_i|}=0.$$ Indeed, if it's nonzero, we can find $(x_i)$ such that $$\limsup\frac{\left|\sum_{y\in F_ix_i}f(y)\right|}{|F_i|}>0;$$ then the uniform measure on $F_ix_i$ accumulates to an invariant mean that is nonzero at $f$. The converse follows from the fact that $$f_i(x)=\sum_{y\in F_ix}\frac{f(y)}{|F_i|}$$ has the same mean as $f$ for every invariant mean.