Let $G$ be a countable amenable group and $f\in\ell^\infty(G)$. Denote by $L,R,I$ respetively the sets of left-, right- and bi-invariant means on $G$. Denote by $M_L(f)$ (resp. $M_R(f),M_I(f)$) be the sets of values attained by the integral $\int f(x)d\mu(x)$, when $\mu$ goes over $L$ (resp. $R,I$).
Question: Is it always true that $M_L(f)=M_R(f)=M_I(f)$? Counterexample?
Thanks in advance,