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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The regular representations of measure groupoids


Author: Peter Hahn
Journal: Trans. Amer. Math. Soc. 242 (1978), 35-72
MSC: Primary 46L10; Secondary 28D99, 46K15
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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1978-0496797-8
PII: S 0002-9947(1978)0496797-8
Article copyright: © Copyright 1978 American Mathematical Society