Take the 2-minute tour ×  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,
share|improve this question Write $G^{op}$ for $G$ with the
multiplication reversed. Then $M_L(G^{op}) = M_R(G)$. But
$G^{op} \cong G$ by $x \mapsto x^{-1}$, so $M_L(G) = M_R(G)$. Or
am I overlooking something? –  Tom Leinster Jan 23 '12 at 19:23
@Tom: You are overlooking the fact that these sets depend on a
given function $f$. –  Alain Valette Jan 23 '12 at 19:32 Oh,
silly me. Thanks. –  Tom Leinster Jan 23 '12 at 19:40 I'll edit
the post to clarify that those sets depend on $f$. –  Valerio
Capraro Jan 23 '12 at 19:56 That's very nice of you, but I think
it was perfectly clear before; I was ju