Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-27T13:31:24.632Z Has data issue: false hasContentIssue false
  • English
  • Français

Approximately transitive flows and ITPFI factors

Published online by Cambridge University Press:  19 September 2008

A. Connes  and
E. J. Woods
A. Connes
Affiliation:
Institute des Hautes Études Scientifiques, 35, route de Chartres, 91440 Bures-sur-Yvette, France
E. J. Woods
Affiliation:
Department of Mathematics and Statistics, Queen's University, Kingston, Ontario, K7L 3N6, Canada
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We define a new property of a Borel group action on a Lebesgue measure space, which we call approximate transitivity. Our main results are (i) a type III0 hyperfinite factor is ITPFI if and only if its flow of weights is approximately transitive, and (ii) for ergodic transformations preserving a finite measure, approximate transitivity implies zero entropy.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 5 , Issue 2 , June 1985 , pp. 203 - 236
Copyright
Copyright © Cambridge University Press 1985

References

REFERENCES

[1]Araki, H. & Woods, E. J.. A classification of factors. Publ. Res. Inst. Math. Sci. Ser. A. 4 (1968), 51–130.CrossRefGoogle Scholar
[2]Connes, A.. Une classification des facteurs de type III. Ann. Sci. École Norm. Sup. (4) 6 (1973), 133–252.CrossRefGoogle Scholar
[3]Connes, A.. On hyperfinite factors of type III and Krieger's factors. J. Funct. Anal. 18 (1975), 318–327.CrossRefGoogle Scholar
[4]Connes, A.. Classification of injective factors. Ann. of Math. 104 (1976), 73–115.CrossRefGoogle Scholar
[5]Connes, A. & Takesaki, M.. The flow of weights on factors of type III. Tohoku Math. J. 29 (1977), 473–575.CrossRefGoogle Scholar
[6]Connes, A. & Woods, E. J.. A construction of approximately finite-dimensional non-ITPFI factors. Can. Math. Bull. 23 (1980), 227–230.CrossRefGoogle Scholar
[7]Dixmier, J.. Les Algèbres d'Opérateurs dans l'Espace Hilbertien, 2e éd. Gauthier-Villars: Paris, 1969.Google Scholar
[8]Elliott, G. & Woods, E. J.. The equivalence of various definitions for a properly infinite von Neumann algebra to be approximately finite dimensional. Proc. Amer. Math. Soc. 60 (1976), 175–178.CrossRefGoogle Scholar
[9]Friedman, N.. Introduction to Ergodic Theory. van Nostrand: New York, 1970.Google Scholar
[10]Hawkins, J. & Woods, E. J.. Approximately transitive diffeomorphism of the circle. Proc. Amer. Math. Soc. 90 (1984), 258–262.CrossRefGoogle Scholar
[11]del Junco, A.. Transformations with discrete spectrum are stacking transformations. Can. J. Math. 28 (1976), 836–839.CrossRefGoogle Scholar
[12]Krieger, W.. On the infinite product construction of non-singular transformations of a measure space. Invent. Math. 15 (1972), 144–163.CrossRefGoogle Scholar
[13]Krieger, W.. On ergodic flows and the isomorphism of factors. Math. Ann. 223 (1976), 19–70.CrossRefGoogle Scholar
[14]Lance, C.. Martingale convergence in von Neumann algebras. Math. Proc. Camb. Phil. Soc. 84 (1978), 47–56.CrossRefGoogle Scholar
[15]Murray, F. J. & von Neumann, J.. On rings of operators IV. Ann. of Math. (2) 44 (1943), 716–808.CrossRefGoogle Scholar
[16]Powers, R.. Representations of uniformly hyperfinite algebras and their associated von Neumann rings. Ann. of Math. 86 (1967), 138–171.CrossRefGoogle Scholar
[17]Sakai, S.. C*-algebras and W*-algebras. Springer-Verlag: New York, 1971.Google Scholar
[18]Smorodinsky, M.. Ergodic Theory, Entropy. Lecture Notes in Maths. 214. Springer-Verlag: New York, 1971.CrossRefGoogle Scholar
[19]Størmer, E.. Hyperfinite product factors, I, II, III. Arkiv för matematik 9 (1971), 165–170;CrossRefGoogle Scholar
J. Funct. Anal. 10 (1972), 471–481;CrossRefGoogle Scholar
Amer. J. Math. 97 (1975), 589–595.CrossRefGoogle Scholar
[20]Stratila, S.. Modular Theory in von Neumann Algebras. Abacus Press: Tunbridge Wells, 1981.Google Scholar
[21]Woods, E. J.. ITPFI factors—a survey. Proc. of Symposia in Pure Mathematics 38 (1982), Part 2, 25–41.CrossRefGoogle Scholar