  In this work we apply the Dispersive Matrix (DM) method of Refs. [1,2] to the
lattice computations of the Form Factors (FFs) entering the semileptonic $B \to
D^* \ell \nu_\ell$ decays, recently produced by the FNAL/MILC Collaborations
[3] at small, but non-vanishing values of the recoil variable ($w-1$). Thanks
to the DM method we obtain the FFs in the whole kinematical range accessible to
the decay in a completely model-independent and non-perturbative way,
implementing exactly both unitarity and kinematical constraints. Using our
theoretical bands of the FFs we extract $\vert V_{cb} \vert$ from the
experimental data and compute the theoretical value of $R(D^*)$. Our final
result for $\vert V_{cb} \vert$ reads $\vert V_{cb} \vert = (41.3 \pm 1.7)
\cdot 10^{-3}$, compatible with the most recent inclusive estimate at the
$0.5\sigma$ level. Moreover, we obtain the pure theoretical value $R(D^*) =
0.275 \pm 0.008$, which is compatible with the experimental world average at
the $\sim 1.3 \sigma$ level.
