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. $R,I$).



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> **Question:** Is it always true that $M\_L(f)=M\_R(f)=M\_I(f)$? Counterexample?
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Thanks in advance,


Valerio



Left mean values vs right mean values
For a given amenable group $G$, these sets will coincide for all $f \in \ell^\infty(G)$ if and only if the sets $L$, $R$, and $I$ coincide. Just notice that if $f \in \ell^\infty(G)$, and $g \in G$ then we have $M\_L( f - \lambda\_g(f) ) = \{ 0 \}$.


