J Austral Math Soc Ser A 45 pp249--274, 1988.

Cartan Subalgebras of Regular Extensions of von Neumann Algebras

Colin E. Sutherland

Abstract

We analyze the structure of a regular extension M ×_g,u Q of a von Neumann algebra M by an action (modulo inner automorphism) g of a discrete group Q, and a non-abelian 2-cycle u for g, under the assumption that the "action" g of Q is cocycle conjugate to an "action" a which leaves globally invariant a Cartan subalgebra C of M. We show that M ×_g,u Q is isomorphic with the algebra of the left regular projective representation of a certain discrete, non-principal groupoid R Ú Q determined by the action of Q on the given Cartan subalgebra, where R is the Takesaki relation associated to the pair (M, C). We apply this description to give a decomposition of the regular representation of a group G into irreducibles, where G is a split extension of a type I group K by an abelian group Q, and work out the details of the author's earlier abstract Plancherel theorem in the case when K is abelian.

1980 AMS Subject Classification: 46L10, 22D25

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Authors

Colin E. Sutherland

Mathematics Department, The University of New South Wales, P.O. Box 1, Kensington, N.S.W. 2033, Australia.