  The present paper aims to understand separability and entanglement in tensor
cones, in the sense of Namioka and Phelps, that arise from the base cones of
operator system tensor products. Of particular interest here are the Toeplitz
and Fej\'er-Riesz operator systems, which are, respectively, operator systems
of Toeplitz matrices and Laurent polynomials (that is, trigonometric
polynomials), and which are related in the operator system category through
duality. Some notable categorical relationships established in this paper are
the C$^*$-nuclearity of Toeplitz and Fej\'er-Riesz operator systems, as well as
their unique operator system structures when tensoring with injective operator
systems. Among the results of this study are two of independent interest: (i) a
matrix criterion, similar to the one involving the Choi matrix, for a linear
map of the Fej\'er-Riesz operator system to be completely positive; (ii) a
completely positive extension theorem for positive linear maps of $n\times n$
Toeplitz matrices into arbritary von Neumann algebras, thereby showing that a
similar extension theorem of Haagerup for $2\times 2$ Toeplitz matrices holds
for Toeplitz matrices of higher dimension.
