Given a manifold B with conical singularities, we consider the cone algebra
with discrete asymptotics, introduced by Schulze, on a suitable scale of L
-p-Sobolev spaces. Ellipticity is proven to be equivalent to the Fredholm p
roperty in these spaces; it turns out to be independent of the choice of p.
We then show that the cone algebra is closed under inversion: whenever an
operator is invertible between the associated Sobolev spaces, its inverse b
elongs to the calculus. We use these results to analyze the behaviour of th
ese operators on L-p(B).