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[Added 10/3/2012] Well, well, well: now I'm in the position of having announced a proof that $F$ is amenable only to have an error be found. The error was finally found by Azer Akhemedov after being overlooked for roughly 4 weeks by myself and 9 or more people who had checked the proof and found no problems. The basic strategy of the proof still may be valid: it began by considering an extension of the free binary system $(\mathbb{T},*)$ on one generator to the finitely additive probability measures on this system: $$\mu * \nu (E) = \int \int \chi_E(s * t) d \nu (t) d \mu (s).$$ It was shown (correctly) that any idempotent measure is $F$-invariant (there is a natural way of identifying $\mathbb{T}$ with the positive elements of $F$). The difficulty came in constructing the idempotent measure. A version of the Kakutani Fixed Point Theorem was used to construct approximations $K_{\mathcal{B},k,n}$ to the set of idempotent measures. The difficulty came in attempting to intersect these compact families of measures. In the proof, it was claimed that the parameter $k$ could be stablized along the an ultrafilter (Lemma 4.13 in the most recent version), allowing one to take a directed intersection of nonempty compact sets. This lemma is likely false and at least is not proved as claimed. One may still be able to argue that a relevant intersection of these approximations is nonempty and hence that there is an idempotent. This seems to require new ideas though.

[Added 10/3/2012] Well, well, well: now I'm in the position of having announced a proof that $F$ is amenable only to have an error be found. The error was finally found by Azer Akhemedov after being overlooked for roughly 4 weeks by myself and 9 or more people who had checked the proof and found no problems. The basic strategy of the proof still may be valid: it began by considering an extension of the free binary system $(\mathbb{T},*)$ on one generator to the finitely additive probability measures on this system: $$\mu * \nu (E) = \int \int \chi_E(s * t) d \nu (t) d \mu (s).$$ It was shown (correctly) that any idempotent measure is $F$-invariant (there is a natural way of identifying $\mathbb{T}$ with the positive elements of $F$). The difficulty came in constructing the idempotent measure. A version of the Kakutani Fixed Point Theorem was used to construct approximations $K_{\mathcal{B},k,n}$ to the set of idempotent measures. The error occurs in attempting to intersect these compact families of measures. In the proof, it was claimed that the parameter $k$ could be stablized along the an ultrafilter (Lemma 4.13 in the most recent version), allowing one to take a directed intersection of nonempty compact sets. This lemma is likely false and at least is not proved as claimed. One may still be able to argue that a relevant intersection of these approximations is nonempty and hence that there is an idempotent. This seems to require new ideas though.

[Added 10/3/2012] Well, well, well: now I'm in the position of having announced a proof that $F$ is amenable only to have an error be found. The error was finally found by Azer Akhemedov after being overlooked for roughly 4 weeks by myself and 9 or more people who had checked the proof and found no problems. The basic strategy of the proof still may be valid: it began by considering an extension of the free binary system $(\mathbb{T},*)$ on one generator to the finitely additive probability measures on this system: $$\mu * \nu (E) = \int \int \chi_E(s * t) d \nu (t) d \mu (s).$$ It was shown (correctly) that any idempotent measure is $F$-invariant (there is a natural way of identifying $\mathbb{T}$ with the positive elements of $F$). The difficulty came in constructing the idempotent measure. A version of the Kakutani Fixed Point Theorem was used to construct approximations $K_{\mathcal{B},k,n}$ to the set of idempotent measures. The difficulty came in attempting to intersect these compact families of measures. In the proof, it was claimed that the parameter $k$ could be stablized along the an ultrafilter (Lemma 4.13 in the most recent version), allowing one to take a directed intersection of nonempty compact sets. This lemma is likely false and at least is not proved as claimed. One may still be able to argue that a relevant intersection of these approximations is nonempty and hence that there is an idempotent. This seems to require new ideas though.

[Added 2/3/2011] Details on what is incorrect with Shavgulidze's proof of the amenablity of $F$ can be found here.