Geometry of a crossed product
HTML articles powered by AMS MathViewer
- by Igor Nikolaev PDF
- Proc. Amer. Math. Soc. 128 (2000), 1177-1183 Request permission
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
- 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, DOI 10.1090/mmono/153
- Kenneth R. Davidson, $C^*$-algebras by example, Fields Institute Monographs, vol. 6, American Mathematical Society, Providence, RI, 1996. MR 1402012, DOI 10.1090/fim/006
- Edward G. Effros and Chao Liang Shen, Approximately finite $C^{\ast }$-algebras and continued fractions, Indiana Univ. Math. J. 29 (1980), no. 2, 191–204. MR 563206, DOI 10.1512/iumj.1980.29.29013
- 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 397420, DOI 10.1016/0021-8693(76)90242-8
- 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
- E. Hopf, Ergodentheorie, in: Ergebnisse der Math. und ihrer Grenzgebiete, Bd.5, Springer 1970.
- A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems. (Encyclopedia of mathematics and its applications). Cambridge Univ. Press, 1995.
- H. Minkowski, Geometrie der Zahlen, Leipzig, 1910.
- P. J. Myrberg, Ein Approximationssatz fur die Fuchsschen Gruppen, Acta Math. 57 (1931), 389-409.
- I. Nikolaev, Artin’s numbers, CRM-2534, Univ. de Montréal, Preprint (1998); available http://www.crm.umontreal.ca
- 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, DOI 10.2140/pjm.1989.136.329
- A. Weil, Les familles de courbes sur le tore. Mat. Sbornik 1 (1936), No 5, 779-781.
Additional Information
- Igor Nikolaev
- Affiliation: CRM, Université de Montréal, Montréal H3C 3J7, Canada; Fields Institute, 222 College Stree, Toronto, Canada M5T 3J1
- Email: nikolaev@crm.umontreal.ca
- 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
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 1177-1183
- MSC (1991): Primary 46L40, 57R30, 58F10
- DOI: https://doi.org/10.1090/S0002-9939-99-05253-3
- MathSciNet review: 1654101