Transactions of the American Mathematical Society

Author(s): Valentin Deaconu; Fred Shultz
Journal: Trans. Amer. Math. Soc. 359 (2007), 1889-1924.
MSC (2000): Primary 46L80; Secondary 37E05
Posted: November 22, 2006
Retrieve article in: PDF DVI PostScript

Abstract: For each piecewise monotonic map $\tau$ of

, we associate a pair of C*-algebras $F_\tau$ and $O_\tau$ and calculate their K-groups. The algebra $F_\tau$ is an AI-algebra. We characterize when $F_\tau$ and $O_\tau$ are simple. In those cases, $F_\tau$ has a unique trace, and $O_\tau$ is purely infinite with a unique KMS state. In the case that $\tau$ is Markov, these algebras include the Cuntz-Krieger algebras

, and the associated AF-algebras

. Other examples for which the K-groups are computed include tent maps, quadratic maps, multimodal maps, interval exchange maps, and $\beta$ -transformations. For the case of interval exchange maps and of $\beta$ -transformations, the C*-algebra $O_\tau$ coincides with the algebras defined by Putnam and Katayama-Matsumoto-Watatani, respectively.

1.: C. Anantharaman-Delaroche, Purely infinite $C\sp *$ -algebras arising from dynamical systems. Bull. Soc. Math. France 125 (1997), no. 2, 199-225.MR 1478030 (99i:46051)
2.: V. Arzumanian and J. Renault, Examples of pseudogroups and their $C\sp *$ -algebras. Operator algebras and quantum field theory (Rome, 1996), 93-104, Internat. Press, Cambridge, MA, 1997. MR 1491110 (99a:46101)
3.: V. A. Arzumanian and A. M. Vershik, Star algebras associated with endomorphisms. Operator algebras and group representations, Vol. I (Neptun, 1980), 17-27, Monogr. Stud. Math., 17, Pitman, Boston, MA, 1984.MR 0731760 (85k:46063)
4.: B. Blackadar, Symmetries of the CAR algebra. Ann. of Math. (2) 131 (1990), no. 3, 589-623. MR 1053492 (91i:46084)
5.: B. Brenken, C*-algebras associated with topological relations, J. Ramanujan Math. Soc. 19 (2004), no. 1, 35-55. MR 2054608 (2004m:46134)
6.: L. S. Block and W. A. Coppel, Dynamics in one dimension. Lecture Notes in Mathematics, 1513. Springer-Verlag, Berlin, 1992.MR 1176513 (93g:58091)
7.: M. Boyle, D. Fiebig, and U. Fiebig, A dimension group for local homeomorphisms and endomorphisms of one-sided shifts of finite type. J. Reine Angew. Math. 487 (1997), 27-59.MR 1454258 (98i:54020)
8.: P. Collet and J. Eckmann, Iterated maps on the interval as dynamical systems, Progress in Physics 1, Birkhäuser, Boston, 1980.MR 0613981 (82j:58078)
9.: J. Cuntz, A class of $C\sp{*}$ -algebras and topological Markov chains. II. Reducible chains and the Ext-functor for $C\sp{*}$ -algebras. Invent. Math. 63 (1981), no. 1, 25-40. MR 0608527 (82f:46073b)
10.: J. Cuntz and W. Krieger, A class of C*-algebras and topological Markov chains. Invent. Math. 56 (1980), no. 3, 251-268. MR 0561974 (82f:46073a)
11.: K. Davidson, C*-algebras by example. Fields Institute Monographs, 6. American Mathematical Society, Providence, RI, 1996. MR 1402012 (97i:46095)
12.: V. Deaconu, Groupoids associated with endomorphisms. Trans. Amer. Math. Soc. 347 (1995), no. 5, 1779-1786. MR 1233967 (95h:46104)
13.: V. Deaconu and P. Muhly, C*-algebras associated with branched coverings. Proc. Amer. Math. Soc. 129 (2001), no. 4, 1077-1086 (electronic).MR 1814145 (2002a:46075)
14.: V. Deaconu and F. Shultz, C*-algebras associated with interval maps, 40 pages, arXiv:math. OA/0405469.
15.: M. Denker, and M. Urbanski, On the existence of conformal measures. Trans. Amer. Math. Soc. 328 (1991), no. 2, 563-587.MR 1014246 (92k:58155)
16.: E. Effros, Dimensions and C*-algebras. CBMS Regional Conference Series in Mathematics, 46. Conference Board of the Mathematical Sciences, Washington, D.C., 1981. MR 0623762 (84k:46042)
17.: E. G. Effros, D. E. Handelman, and C. L. Shen, Dimension groups and their affine representations. Amer. J. Math. 102 (1980), no. 2, 385-407.MR 0564479 (83g:46061)
18.: G. 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)
19.: G. Elliott, On the classification of C*-algebras of real rank zero, J. Reine Angew. Math. 443 (1993), 179-219. MR 1241132 (94i:46074)
20.: R. Exel, personal communication.
21.: R. Exel and A. Vershik, C*-algebras of irreversible dynamical systems. Canad. J. Math. 58 (2006), no. 1, 39-63. MR 2195591
22.: T. Giordano, I. Putnam, and C. Skau, Topological orbit equivalence and C*-crossed products. J. Reine Angew. Math. 469 (1995), 51-111. MR 1363826 (97g:46085)
23.: K. R. Goodearl, Partially ordered abelian groups with interpolation. Mathematical Surveys and Monographs, 20. American Mathematical Society, Providence, RI, 1986. MR 0845783 (88f:06013)
24.: F. Hofbauer, The structure of piecewise monotonic transformations. Ergodic Theory Dynamical Systems 1 (1981), no. 2, 159-178. MR 0661817 (83k:58065)
25.: L. Jonker and D. Rand, Bifurcations in one dimension. I. The nonwandering set. Invent. Math. 62 (1981), no. 3, 347-365.MR 0604832 (83d:58057a)
26.: T. Kajiwara and Y. Watatani, C*-algebras associated with complex dynamical systems. Indiana Univ. Math. J. 54 (2005), no. 3, 755-778. MR 2151233 (2006b:46074)
27.: Y. Katayama, K. Matsumoto, and Y. Watatani, Simple $C\sp *$ -algebras arising from $\beta$ -expansion of real numbers. Ergodic Theory Dynam. Systems 18 (1998), no. 4, 937-962. MR 1645334 (99m:46136)
28.: T. Katsura, A class of C*-algebras generalizing both graph algebras and homeomorphism C*-algebras I, fundamental results. Trans. Amer. Math. Soc. 356 (2004), no. 11, 4287-4322. MR 2067120 (2005b:46119)
29.: T. Katsura, On C*-algebras associated with C*-correspondences, J. Funct. Anal. 217 (2004), no. 2, 366-401. MR 2102572 (2005e:46099)
30.: M. Keane, Interval exchange transformations. Math. Z. 141 (1975), 25-31. MR 0357739 (50:10207)
31.: M. Keane, Non-ergodic interval exchange transformations. Israel J. Math. 26 (1977), no. 2, 188-196.MR 0435353 (55:8313)
32.: H. Keynes and D. Newton, A ``minimal", non-uniquely ergodic interval exchange transformation. Math. Z. 148 (1976), no. 2, 101-105. MR 0409766 (53:13518)
33.: E. Kirchberg, The classification of purely infinite C*-algebras using Kasparov's theory, to appear in the Fields Institute Communication series.
34.: W. Krieger, On a dimension for a class of homeomorphism groups. Math. Ann. 252 (1979/80), no. 2, 87-95.MR 0593623 (82b:46083)
35.: W. Krieger, On dimension functions and topological Markov chains. Invent. Math. 56 (1980), no. 3, 239-250.MR 0561973 (81m:28018)
36.: A. Kumjian, Preliminary algebras arising from local homeomorphisms. Math. Scand. 52 (1983), no. 2, 269-278. MR 0702958 (85b:46078)
37.: A. Kumjian, Fell bundles over groupoids. Proc. Amer. Math. Soc. 126 (1998), no. 4, 1115-1125. MR 1443836 (98i:46055)
38.: A. Kumjian, D. Pask, and I. Raeburn, Cuntz-Krieger algebras of directed graphs. Pacific J. Math. 184 (1998), no. 1, 161-174. MR 1626528 (99i:46049)
39.: A. Kumjian and J. Renault, KMS states on C*-algebras associated to expansive maps, Proc. Amer. Math. Soc. 134 (2006), no. 7, 2067-2078.MR 2215776
40.: N. Martins, R. Severino, and J. S. Ramos, -theory for Cuntz-Krieger algebras arising from real quadratic maps. Int. J. Math. Math. Sci. 2003, no. 34, 2139-2146. MR 1991959 (2004g:46085)
41.: P. S. Muhly, J. N. Renault, and D. P. Williams, Equivalence and isomorphism for groupoid $C\sp *$ -algebras. J. Operator Theory 17 (1987), no. 1, 3-22. MR 0873460 (88h:46123)
42.: M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings. Studia Math. 67 (1980), no. 1, 45-63.MR 0579440 (82a:58030)
43.: D. Lind and B. Marcus, An introduction to symbolic dynamics and coding. Cambridge University Press, Cambridge, 1995. MR 1369092 (97a:58050)
44.: W. de Melo and S. van Strien, One-dimensional dynamics. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 25. Springer-Verlag, Berlin, 1993. MR 1239171 (95a:58035)
45.: M. Newman, Integral Matrices, Pure and Applied Mathematics, Vol. 45. Academic Press, New York, London, 1972. MR 0340283 (49:5038)
46.: W. Parry, Symbolic dynamics and transformations of the unit interval. Trans. Amer. Math. Soc. 122 (1966), 368-378.MR 0197683 (33:5846)
47.: W. Parry and S. Tuncel, Classification problems in ergodic theory. London Mathematical Society Lecture Note Series, 67. Statistics: Textbooks and Monographs, 41. Cambridge University Press, Cambridge, New York, 1982. MR 0666871 (84g:28024)
48.: W. L. Paschke, The crossed product of a $C\sp{*}$ -algebra by an endomorphism. Proc. Amer. Math. Soc. 80 (1980), no. 1, 113-118.MR 0574518 (81m:46081)
49.: W. L. Paschke, -theory for actions of the circle group on $C\sp{*}$ -algebras. J. Operator Theory 6 (1981), no. 1, 125-133.MR 0637006 (82m:46074)
50.: N. C. Phillips, A classification theorem for nuclear purely infinite simple $C\sp *$ -algebras. Doc. Math. 5 (2000), 49-114 (electronic). MR 1745197 (2001d:46086b)
51.: M. Pimsner, A class of C*-algebras generalizing both Cuntz-Krieger algebras and crossed products by $\mathbb{Z}$ . Free probability theory (Waterloo, ON, 1995), 189-212, Fields Inst. Commun., 12, Amer. Math. Soc., Providence, RI, 1997. MR 1426840 (97k:46069)
52.: M. Pimsner and D. Voiculescu, -groups of reduced crossed products by free groups. J. Operator Theory 8 (1982), no. 1, 131-156. MR 0670181 (84d:46092)
53.: Y. T. Poon, A -theoretic invariant for dynamical systems, Trans. Amer. Math. Soc. 311 (1989), 515-533.MR 0978367 (90c:46091)
54.: C. Preston, Iterates of piecewise monotone mappings on an interval. Lecture Notes in Mathematics, 1347. Springer-Verlag, Berlin, 1988.MR 0969131 (89m:58109)
55.: I. Putnam, The C*-algebras associated with minimal homeomorphisms of the Cantor set, Pacific J. Math. 136 (1989), no. 2, 329-353.MR 0978619 (90a:46184)
56.: I. Putnam, C*-algebras arising from interval exchange transformations. J. Operator Theory 27 (1992), no. 2, 231-250. MR 1249645 (95a:46090)
57.: I. Putnam, K. Schmidt, and C. Skau, C*-algebras associated with Denjoy homeomorphisms of the circle. J. Operator Theory 16 (1986), no. 1, 99-126. MR 0847334 (87j:46126)
58.: J. Renault, A groupoid approach to $C\sp{*}$ -algebras. Lecture Notes in Mathematics, 793. Springer, Berlin, 1980.MR 0584266 (82h:46075)
59.: J. Renault, Cuntz-like Algebras, Operator theoretical methods (Timisoara, 1998), Theta Found., Bucharest, 2000, pp. 371-386. MR 1770333 (2001g:46130)
60.: J. Renault, The Radon-Nikodym problem for approximately proper equivalence relations, Ergodic Theory Dynam. Systems 25 (2005), no. 5, 1643-1672. MR 2173437 (2006h:46065)
61.: M. Rørdam, F. Larsen, and N. J. Lausten, An Introduction to K-theory for C*-algebras. London Mathematical Society Student Texts 49, Cambridge University Press, Cambridge, 2000. MR 1783408 (2001g:46001)
62.: D. Ruelle, Dynamical zeta functions for piecewise monotone maps of the interval. CRM Monograph Series, 4. American Mathematical Society, Providence, RI, 1994. MR 1274046 (95m:58101)
63.: F. Shultz, Dimension groups for interval maps, New York J. Math. 11 (2005), 477-517. MR 2188252 (2006g:37062)
64.: F. Shultz, Dimension groups for interval maps II: The transitive case, Ergodic Theory Dynamical Systems, to appear. arXiv:math.DS/0405467.
65.: J. Spielberg, Cuntz-Krieger algebras associated with Fuchsian groups. Ergodic Theory Dynam. Systems 13 (1993), no. 3, 581-595. MR 1245830 (94k:46143)
66.: P. Walters, Equilibrium states for $\beta$ -transformations and related transformations, Math. Z. 159 (1978), 65-88. MR 0466492 (57:6370)
67.: P. Walters, An introduction to ergodic theory. Graduate Texts in Mathematics, 79. Springer-Verlag, New York, Berlin, 1982.MR 0648108 (84e:28017)

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 46L80, 37E05

Valentin Deaconu
Affiliation: Department of Mathematics, University of Nevada, Reno, Nevada 89557
Email: vdeaconu@unr.edu

Fred Shultz
Affiliation: Department of Mathematics, Wellesley College, Wellesley, Massachusetts 02481
Email: fshultz@wellesley.edu

DOI: 10.1090/S0002-9947-06-04112-2
PII: S 0002-9947(06)04112-2
Keywords: Dimension group, interval map, piecewise monotonic, unimodal map, tent map, Markov map, $\beta$-shift, interval exchange map, C*-algebra, Cuntz-Krieger algebra, Matsumoto algebra
Received by editor(s): August 1, 2004
Received by editor(s) in revised form: June 11, 2005
Posted: November 22, 2006
Copyright of article: Copyright 2006, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS