The regular representations of measure groupoids

Abstract

Techniques are developed to study the regular representation and σ \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 σ \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 σ \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 σ \sigma . To treat nonergodic groupoids, an ergodic decomposition theorem is provided.