  We define a class of non-compact Fano toric manifolds, called admissible
toric manifolds, for which Floer theory and quantum cohomology are defined. The
class includes Fano toric negative line bundles, and it allows blow-ups along
fixed point sets. We prove closed-string mirror symmetry for this class of
manifolds: the Jacobian ring of the superpotential is the symplectic cohomology
(not the quantum cohomology). Moreover, SH(M) is obtained from QH(M) by
localizing at the toric divisors. We give explicit presentations of SH(M) and
QH(M), using ideas of Batyrev, McDuff and Tolman. Assuming that the
superpotential is Morse (or a milder semisimplicity assumption), we prove that
the wrapped Fukaya category for this class of manifolds satisfies the toric
generation criterion, i.e. is split-generated by the natural Lagrangian torus
fibres of the moment map with suitable holonomies. In particular, the wrapped
category is compactly generated and cohomologically finite. The proof uses a
deformation argument, via a generic generation theorem and an argument about
continuity of eigenspaces. We also prove that for any closed Fano toric
manifold, if the superpotential is Morse (or a milder semisimplicity
assumption) then the Fukaya category satisfies the toric generation criterion.
The key ingredients are non-vanishing results for the open-closed string map,
using tools from the paper by Ritter-Smith (we also prove a conjecture from
that paper that any monotone toric negative line bundle contains a
non-displaceable monotone Lagrangian torus). We also need to extend the class
of Hamiltonians for which the maximum principle holds for symplectic manifolds
conical at infinity, thus extending the class of Hamiltonian circle actions for
which invertible elements can be constructed in SH(M).
