Let (G,Z) be a second
countable locally compact topological transformation group, 𝒰(G,Z) the
associated C∗-algebra and L a certain naturally constructed representation of
𝒰(G,Z) on L2(G × Z,dg × dα),dg being left Haar measure on G and α a
quasi-invariant ergodic probability measure on Z. Representations of 𝒰(G,Z)
constructed from positive-definite measures on G × Z are used to prove
that 𝒰(G,Z) is type I if and only if all the isotropy subgroups are type I
and Z∕G is T0, and, under the assumption of a common central isotropy
subgroup, that L has no type I component if α is nontransitive. By means of
quasi-unitary algebras, necessary and sufficient conditions are derived for L to be
semi-finite under the weaker assumption of a common type I unimodular isotropy
subgroup.
