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
Z∕G, 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 Hilbert 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.
