The regular representations of measure groupoids

Hahn, Peter

doi:10.1090/S0002-9947-1978-0496797-8

The regular representations of measure groupoids
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Trans. Amer. Math. Soc. 242 (1978), 35-72 Request permission

Abstract:

Techniques are developed to study the regular representation and $\sigma$-regular representations of measure groupoids. Convolution, involution, a modular Hilbert algebra, and local and global versions of the regular representation are defined. The associated von Neumann algebras, each uniquely determined by the groupoid and the cocycle $\sigma$, provide a generalization of the group-measure space construction. When the groupoid is principal and ergodic, these algebras are factors. Necessary and sufficient conditions for the $\sigma$-regular representations of a principal ergodic groupoid to be of type I, II, or III are given, as well as a description of the flow of weights; these are independent of $\sigma$. To treat nonergodic groupoids, an ergodic decomposition theorem is provided.

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