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Transactions of the American Mathematical Society

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The regular representations of measure groupoids


Author: Peter Hahn
Journal: Trans. Amer. Math. Soc. 242 (1978), 35-72
MSC: Primary 46L10; Secondary 28D99, 46K15
DOI: https://doi.org/10.1090/S0002-9947-1978-0496797-8
MathSciNet review: 496797
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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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DOI: https://doi.org/10.1090/S0002-9947-1978-0496797-8
Article copyright: © Copyright 1978 American Mathematical Society

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