  The author's recent results on spectral invariant dense subalgebras of
C*-algebras associated with dynamical systems are summarized. If G is a
compactly generated polynomial growth Type R Lie group, and the action of G on
S(M) (Schwartz functions on a locally compact G-space M) is tempered in a
certain sense, then there is a natural smooth crossed product S(G X M) which is
dense and spectral invariant in the C*-crossed product C*(G X M).
