  We define the notion of strong spectral invariance for a dense Frechet
subalgebra A of a Banach algebra B. We show that if A is strongly spectral
invariant in a C*-algebra B, and G is a compactly generated polynomial growth
Type R Lie group, not necessarily connected, then the smooth crossed product
G\rtimes A is spectral invariant in the C*-crossed product G\rtimes B. Examples
of such groups are given by finitely generated polynomial growth discrete
groups, compact or connected nilpotent Lie groups, the group of Euclidean
motions on the plane, the Mautner group, or any closed subgroup of one of
these. Our theorem gives the spectral invariance of G\rtimes A if A is the set
of C^{\infty}-vectors for the action of G on B, or if B= C_{0}(M), and A is a
set of G-differentiable Schwartz functions S(M) on M. This gives many examples
of spectral invariant dense subalgebras for the C*-algebras associated with
dynamical systems. We also obtain relevant results about exact sequences,
subalgebras, tensoring by smooth compact operators, and strong spectral
invariance in L_{1}(G, B).
