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Cocycles and continuity

Author: Howard Becker
Journal: Trans. Amer. Math. Soc. 365 (2013), 671-719
MSC (2010): Primary 03E15, 22A25, 28D15, 37A20, 43A65
Published electronically: July 19, 2012
MathSciNet review: 2995370
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Abstract: The topic of this paper is the Mackey Cocycle Theorem: every Borel almost cocycle is equivalent to a Borel strict cocycle. This is a theorem about locally compact groups which is not true for arbitrary Polish groups. We discuss the theorem, the open question of whether the theorem generalizes to some nonlocally compact Polish groups, the generalization to non-Borel cocycles, and other subjects associated with the theorem. Traditionally, the subject of cocycles and related matters has been considered in the context of standard Borel $ G$-spaces. It is now known that a standard Borel $ G$-space has a topological realization as a Polish $ G$-space. This makes it possible to consider the subject from a topological point of view. The main theorem of this paper is that the conclusion of the Mackey Cocycle Theorem is equivalent to continuity properties of the almost cocycle. Even in the locally compact case, this continuity is a new result.

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Additional Information

Howard Becker
Affiliation: PMB 128, 4840 Forest Dr., Ste. 6–B, Columbia, South Carolina 29206-4810

Received by editor(s): March 21, 2010
Received by editor(s) in revised form: February 16, 2011
Published electronically: July 19, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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