  A new proof is presented of a theorem of L.~Gurvits, which states that the
cone of positive block-Toeplitz matrices with matrix entries has no entangled
elements. The proof of the Gurvits separation theorem is achieved by making use
of the structure of the operator system dual of the operator system
$C(S^1)^{(n)}$ of $n\times n$ Toeplitz matrices over the complex field, and by
determining precisely the structure of the generators of the extremal rays of
the positive cones of the operator systems $C(S^1)^{(n)}\omin\B(\H)$ and
$C(S^1)_{(n)}\omin\B(\H)$, where $\H$ is an arbitrary Hilbert space and
$C(S^1)_{(n)}$ is the operator system dual of $C(S^1)^{(n)}$. Our approach also
has the advantage of providing some new information concerning positive
Toeplitz matrices whose entries are from $\B(\H)$ when $\H$ has infinite
dimension. In particular, we prove that normal positive linear maps $\psi$ on
$\B(\H)$ are partially completely positive in the sense that $\psi^{(n)}(x)$ is
positive whenever $x$ is a positive $n\times n$ Toeplitz matrix with entries
from $\B(\H)$. We also establish a certain factorisation theorem for positive
Toeplitz matrices (of operators), showing an equivalence between the Gurvits
approach to separation and an earlier approach of T.~Ando to universality.
