Vol. 48, No. 1, 1973

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The type of some C and W-algebras associated with transformation groups

Elliot Charles Gootman

Vol. 48 (1973), No. 1, 93–106

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 × ),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.

Mathematical Subject Classification 2000
Primary: 22D25
Received: 26 June 1972
Published: 1 September 1973
Elliot Charles Gootman