  We investigate the equivariant intersection cohomology of a toric variety.
Considering the defining fan of the variety as a finite topological space with
the subfans being the open sets (that corresponds to the "toric" topology given
by the invariant open subsets), equivariant intersection cohomology provides a
sheaf (of graded modules over a sheaf of graded rings) on that "fan space". We
prove that this sheaf is a "minimal extension sheaf", i.e., that it satisfies
three relatively simple axioms which are known to characterize such a sheaf up
to isomorphism. In the verification of the second of these axioms, a key role
is played by "equivariantly formal" toric varieties, where equivariant and
"usual" (non-equivariant) intersection cohomology determine each other by
Kunneth type formulae. Minimal extension sheaves can be constructed in a purely
formal way and thus also exist for non-rational fans. As a consequence, we can
extend the notion of an equivariantly formal fan even to this general setup. In
this way, it will be possible to introduce "virtual" intersection cohomology
for equivariantly formal non-rational fans.
