Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


Geometry of a crossed product

Author: Igor Nikolaev
Journal: Proc. Amer. Math. Soc. 128 (2000), 1177-1183
MSC (1991): Primary 46L40, 57R30, 58F10
Published electronically: October 18, 1999
MathSciNet review: 1654101
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We introduce a continuous dimension function $\alpha: \bullet\to\mathbb{R}$ on the Grothendieck group $K_0$ over the crossed product $C^*$-algebra $C(X)\rtimes _{\phi}\mathbb{Z}$. The function $\alpha$ has an elegant geometry: on every minimal flow $\phi^t$ it takes the value of the ``rotation number" of $\phi^t$; such a problem was posed in 1936 by A. Weil.

References [Enhancements On Off] (What's this?)

  • 1. S. Kh. Aranson, G. R. Belitsky, and E. V. Zhuzhoma, Introduction to the qualitative theory of dynamical systems on surfaces, Translations of Mathematical Monographs, vol. 153, American Mathematical Society, Providence, RI, 1996. Translated from the Russian manuscript by H. H. McFaden. MR 1400885 (97c:58135)
  • 2. Kenneth R. Davidson, 𝐶*-algebras by example, Fields Institute Monographs, vol. 6, American Mathematical Society, Providence, RI, 1996. MR 1402012 (97i:46095)
  • 3. Edward G. Effros and Chao Liang Shen, Approximately finite 𝐶*-algebras and continued fractions, Indiana Univ. Math. J. 29 (1980), no. 2, 191–204. MR 563206 (81g:46076),
  • 4. George A. Elliott, On the classification of inductive limits of sequences of semisimple finite-dimensional algebras, J. Algebra 38 (1976), no. 1, 29–44. MR 0397420 (53 #1279)
  • 5. Thierry Giordano, Ian F. Putnam, and Christian F. Skau, Topological orbit equivalence and 𝐶*-crossed products, J. Reine Angew. Math. 469 (1995), 51–111. MR 1363826 (97g:46085)
  • 6. E. Hopf, Ergodentheorie, in: Ergebnisse der Math. und ihrer Grenzgebiete, Bd.5, Springer 1970.
  • 7. A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems. (Encyclopedia of mathematics and its applications). Cambridge Univ. Press, 1995.
  • 8. H. Minkowski, Geometrie der Zahlen, Leipzig, 1910.
  • 9. P. J. Myrberg, Ein Approximationssatz fur die Fuchsschen Gruppen, Acta Math. 57 (1931), 389-409.
  • 10. I. Nikolaev, Artin's numbers, CRM-2534, Univ. de Montréal, Preprint (1998); available
  • 11. Ian F. Putnam, The 𝐶*-algebras associated with minimal homeomorphisms of the Cantor set, Pacific J. Math. 136 (1989), no. 2, 329–353. MR 978619 (90a:46184)
  • 12. A. Weil, Les familles de courbes sur le tore. Mat. Sbornik 1 (1936), No 5, 779-781.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 46L40, 57R30, 58F10

Retrieve articles in all journals with MSC (1991): 46L40, 57R30, 58F10

Additional Information

Igor Nikolaev
Affiliation: CRM, Université de Montréal, Montréal H3C 3J7, Canada; Fields Institute, 222 College Stree, Toronto, Canada M5T 3J1

PII: S 0002-9939(99)05253-3
Keywords: Dimension group, continued fraction, minimal flow
Received by editor(s): November 14, 1997
Received by editor(s) in revised form: June 17, 1998
Published electronically: October 18, 1999
Communicated by: David R. Larson
Article copyright: © Copyright 2000 American Mathematical Society