Cocycles and continuity

Author:
Howard Becker

Journal:
Trans. Amer. Math. Soc. **365** (2013), 671-719

MSC (2010):
Primary 03E15, 22A25, 28D15, 37A20, 43A65

DOI:
https://doi.org/10.1090/S0002-9947-2012-05570-X

Published electronically:
July 19, 2012

MathSciNet review:
2995370

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 -spaces. It is now known that a standard Borel -space has a topological realization as a Polish -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.

**1.**M. Balcerzak, A classification of -ideals in Polish groups,*Demonstratio Mathematica***20**(1987), 77-88. MR**941406 (89e:54076)****2.**H. Becker, Polish group actions: dichotomies and generalized elementary embeddings,*Journal of the American Mathematical Society***11**(1998), 397-449. MR**1478843 (99g:03051)****3.**H. Becker and A.S. Kechris,*The Descriptive Set Theory of Polish Group Actions*, Cambridge University Press, 1996. MR**1425877 (98d:54068)****4.**R.C. Fabec,*Fundamentals of Infinite Dimensional Representation Theory*, Chapman and Hall/CRC, 2000. MR**1775825 (2001m:22001)****5.**F. Fidaleo, Remarks on the imprimitivity theorem for nonlocally compact Polish groups,*Infinite Dimensional Analysis, Quantum Probability and Related Topics***3**(2000), 247-262. MR**1812700 (2002e:22001)****6.**W. Herer and J.P.R. Christensen, On the existence of pathological submeasures and the construction of exotic topological groups,*Math. Ann.***213**(1975), 203-210. MR**0412369 (54:495)****7.**A.S. Kechris,*Classical Descriptive Set Theory*, Springer-Verlag, 1995. MR**1321597 (96e:03057)****8.**G.W. Mackey, Ergodic theory and virtual groups,*Math. Ann.***166**(1966), 187-207. MR**0201562 (34:1444)****9.**G.W. Mackey,*Unitary Group Representations in Physics, Probability and Number Theory*, Addison-Wesley, 1989. MR**1043174 (90m:22002)****10.**D. Montgomery and L. Zippin,*Topological Transformation Groups*, Interscience Publishers, 1955. MR**0073104 (17:383b)****11.**Y.N. Moschovakis,*Descriptive Set Theory*, North-Holland, 1980. MR**561709 (82e:03002)****12.**J.C. Oxtoby,*Measure and Category*, second edition, Springer-Verlag, 1980. MR**584443 (81j:28003)****13.**I. Recław and P. Zakrzewski, Fubini properties of ideals,*Real Analysis Exchange***25**(1999/2000), 565-578. MR**1778511 (2001e:03086)****14.**S. Solecki, Polish group topologies, in*Sets and Proofs*, (S.B. Cooper and J.K. Truss, Eds.), Cambridge University Press, 1999, 339-364. MR**1720580 (2000j:54045)****15.**V.S. Varadarajan,*Geometry of Quantum Theory*, second edition, Springer-Verlag, 1985. MR**805158 (87a:81009)****16.**R. Vaught, Invariant sets in topology and logic,*Fund. Math.***82**(1974), 269-294. MR**0363912 (51:167)****17.**A.M. Vershik, Kolmogorov's example (a survey of actions of infinite-dimensional groups with an invariant probability measure),*Theory of Probability and its Applications***48**(2004), 373-378. MR**2015459 (2004m:37009)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2010):
03E15,
22A25,
28D15,
37A20,
43A65

Retrieve articles in all journals with MSC (2010): 03E15, 22A25, 28D15, 37A20, 43A65

Additional Information

**Howard Becker**

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

Email:
hsbecker@hotmail.com

DOI:
https://doi.org/10.1090/S0002-9947-2012-05570-X

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.