  We recover the Dehornoy order on the braid group $B_{2g+n}$ from the tracial
state on a cluster $C^*$-algebra $\mathbb{A}(S_{g,n})$ associated to the
surface $S_{g,n}$ of genus $g$ with $n$ boundary components. It is proved that
the space of left-ordering of the fundamental group $\pi_1(S_{g,n})$ is a
totally disconnected dense subspace of the projective Teichm\"uller space
$\mathbb{P}T_{g,n}\cong \mathbf{R}^{6g-7+2n}$. In particular, each
left-ordering of $\pi_1(S_{g,n})$ defines the orbit of a Riemann surface
$S_{g,n}$ under the geodesic flow on the space $T_{g,n}$.
