C∗-algebras of transformation groups with smooth orbit space

We investigate the structure of certain locally compact Hausdorff transformation groups (G, X) and the C*-algebras C*(G,X) associated to them. When G and X are second countable and the action is free, we obtain a simple necessary and sufficient condition for C*{G, X) to be a continuous trace algebra, and show that the continuous trace algebras so arising are never " twisted" over their spectra. When G is separable compactly generated Abelian and X contains a totally disconnected set of fixed points whose complement, Z, is a trivial G-principal fiber bundle over its orbit space ZIG, with Z/G compact, C*(G, X) can be described completely using the Brown-Douglas-Fillmore theory of extensions of the compact operators on a separable Hubert space by a commutative algebra. These results yield as special cases the structure of the C*-algebras for several infinite families of solvable locally compact groups.