  We characterize all simple unitarizable representations of the braid group
$B_3$ on complex vector spaces of dimension $d \leq 5$. In particular, we prove
that if $\sigma_1$ and $\sigma_2$ denote the two generating twists of $B_3$,
then a simple representation $\rho:B_3 \to \gl(V)$ (for $\dim V \leq 5$) is
unitarizable if and only if the eigenvalues $\lambda_1, \lambda_2, ...,
\lambda_d$ of $\rho(\sigma_1)$ are distinct, satisfy $|\lambda_i|=1$ and
$\mu^{(d)}_{1i} > 0$ for $2 \leq i \leq d$, where the $\mu^{(d)}_{1i}$ are
functions of the eigenvalues, explicitly described in this paper.
