The type of some<i>C</i>^∗and<i>W</i>^∗-algebras associated with transformation groups

Abstract

Let (G, Z) be a second countable locally compact topological transformation group, ^(G, Z) the associated C*-algebra and L a certain naturally constructed representation of ^(G, Z) on L\G x Z,dg x da), dg being left Haar measure on G and a a quasi-invariant ergodic probability measure on Z. Representations of flf(G, Z) constructed from positive-definite measures on G X Z are used to prove that ^(G, Z) is type I if and only if all the isotropy subgroups are type I and Z/G is o , and, under the assumption of a common central isotropy subgroup, that L has no type / component if a 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 / unimodular isotropy subgroup.