  We investigate spin transport in 2-dimensional insulators, with the long-term
goal of establishing whether any of the transport coefficients corresponds to
the Fu-Kane-Mele index which characterizes 2d time-reversal-symmetric
topological insulators. Inspired by the Kubo theory of charge transport, and by
using a proper definition of the spin current operator, we define the Kubo-like
spin conductance $G_K^{s_z}$ and spin conductivity $\sigma_K^{s_z}$. We prove
that for any gapped, periodic, near-sighted discrete Hamiltonian, the above
quantities are mathematically well-defined and the equality $G_K^{s_z} =
\sigma_K^{s_z}$ holds true. Moreover, we argue that the physically relevant
condition to obtain the equality above is the vanishing of the mesoscopic
average of the spin-torque response, which holds true under our hypotheses on
the Hamiltonian operator. This vanishing condition might be relevant in view of
further extensions of the result, e.g. to ergodic random discrete Hamiltonians
or to Schr\"odinger operators on the continuum. A central role in the proof is
played by the trace per unit volume and by two generalizations of the trace,
the principal value trace and it directional version.
