Suppose that a compact quantum group ${mathcal Q}$ acts faithfully on a smooth, compact, connected manifold $M$, i.e. has a $C^{ast}$ (co)-action $alpha$ on $C(M)$, such that $alpha(C^infty(M)) subseteq C^infty(M, {mathcal Q})$ and the linear span of $alpha(C^infty(M))(1 otimes {mathcal Q})$ is dense in $C^infty(M, {mathcal Q})$ with respect to the Frechet topology. It was conjectured by the author quite a few years ago that ${mathcal Q}$ must be commutative as a $C^{ast}$ algebra i.e. ${mathcal Q} cong C(G)$ for some compact group $G$ acting smoothly on $M$. The goal of this paper is to prove the truth of this conjecture. A remarkable aspect of the proof is the use of probabilistic techniques involving Brownian stopping time.
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