A class of -algebras generalizing both graph algebras and homeomorphism
-algebras I, fundamental results
Author:
Takeshi Katsura
Journal:
Trans. Amer. Math. Soc. 356 (2004), 4287-4322
MSC (2000):
Primary 46L05; Secondary 46L55, 37B99
DOI:
https://doi.org/10.1090/S0002-9947-04-03636-0
Published electronically:
May 28, 2004
MathSciNet review:
2067120
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: We introduce a new class of -algebras, which is a generalization of both graph algebras and homeomorphism
-algebras. This class is very large and also very tractable. We prove the so-called gauge-invariant uniqueness theorem and the Cuntz-Krieger uniqueness theorem, and compute the
-groups of our algebras.
- [AS] R. J. Archbold and J. S. Spielberg, Topologically free actions and ideals in discrete 𝐶*-dynamical systems, Proc. Edinburgh Math. Soc. (2) 37 (1994), no. 1, 119–124. MR 1258035, https://doi.org/10.1017/S0013091500018733
- [AR] Victor Arzumanian and Jean Renault, Examples of pseudogroups and their 𝐶*-algebras, Operator algebras and quantum field theory (Rome, 1996) Int. Press, Cambridge, MA, 1997, pp. 93–104. MR 1491110
- [BHRS]
Bates, T.; Hong, J.; Raeburn, I.; Szymanski, W. The ideal structure of the
-algebras of infinite graphs. Illinois J. Math. 46 (2002), no. 4, 1159-1176.
- [BPRS] Teresa Bates, David Pask, Iain Raeburn, and Wojciech Szymański, The 𝐶*-algebras of row-finite graphs, New York J. Math. 6 (2000), 307–324. MR 1777234
- [CK] Joachim Cuntz and Wolfgang Krieger, A class of 𝐶*-algebras and topological Markov chains, Invent. Math. 56 (1980), no. 3, 251–268. MR 561974, https://doi.org/10.1007/BF01390048
- [D] Valentin Deaconu, Continuous graphs and 𝐶*-algebras, Operator theoretical methods (Timişoara, 1998) Theta Found., Bucharest, 2000, pp. 137–149. MR 1770320
- [DKM] V. Deaconu, A. Kumjian, and P. Muhly, Cohomology of topological graphs and Cuntz-Pimsner algebras, J. Operator Theory 46 (2001), no. 2, 251–264. MR 1870406
- [DT1]
Drinen, D.; Tomforde, M. The
-algebras of arbitrary graphs. To appear in Rocky Mountain J. Math.
- [DT2] D. Drinen and M. Tomforde, Computing 𝐾-theory and 𝐸𝑥𝑡 for graph 𝐶*-algebras, Illinois J. Math. 46 (2002), no. 1, 81–91. MR 1936076
- [DS] Kenneth J. Dykema and Dimitri Shlyakhtenko, Exactness of Cuntz-Pimsner 𝐶*-algebras, Proc. Edinb. Math. Soc. (2) 44 (2001), no. 2, 425–444. MR 1880402, https://doi.org/10.1017/S001309159900125X
- [E] Ruy Exel, A Fredholm operator approach to Morita equivalence, 𝐾-Theory 7 (1993), no. 3, 285–308. MR 1244004, https://doi.org/10.1007/BF00961067
- [EL] Ruy Exel and Marcelo Laca, Cuntz-Krieger algebras for infinite matrices, J. Reine Angew. Math. 512 (1999), 119–172. MR 1703078, https://doi.org/10.1515/crll.1999.051
- [ELQ] Ruy Exel, Marcelo Laca, and John Quigg, Partial dynamical systems and 𝐶*-algebras generated by partial isometries, J. Operator Theory 47 (2002), no. 1, 169–186. MR 1905819
- [FLR] Neal J. Fowler, Marcelo Laca, and Iain Raeburn, The 𝐶*-algebras of infinite graphs, Proc. Amer. Math. Soc. 128 (2000), no. 8, 2319–2327. MR 1670363, https://doi.org/10.1090/S0002-9939-99-05378-2
- [FR] Neal J. Fowler and Iain Raeburn, The Toeplitz algebra of a Hilbert bimodule, Indiana Univ. Math. J. 48 (1999), no. 1, 155–181. MR 1722197, https://doi.org/10.1512/iumj.1999.48.1639
- [HS]
Hong, J. H.; Szymanski, W. The primitive ideal space of the
-algebras of infinite graphs. J. Math. Soc. Japan 56 (2004), no. 1, 45-64.
- [KPW] Tsuyoshi Kajiwara, Claudia Pinzari, and Yasuo Watatani, Ideal structure and simplicity of the 𝐶*-algebras generated by Hilbert bimodules, J. Funct. Anal. 159 (1998), no. 2, 295–322. MR 1658088, https://doi.org/10.1006/jfan.1998.3306
- [Ka1] Katsura, T. The ideal structures of crossed products of Cuntz algebras by quasi-free actions of abelian groups. Canad. J. Math. 55 (2003), no. 6, 1302-1338.
- [Ka2]
Katsura, T. A class of
-algebras generalizing both graph algebras and homeomorphism
-algebras II, examples. Preprint 2004, math.OA/0405268.
- [Ka3]
Katsura, T. A class of
-algebras generalizing both graph algebras and homeomorphism
-algebras III, ideal structures. In preparation.
- [Ka4]
Katsura, T. A class of
-algebras generalizing both graph algebras and homeomorphism
-algebras IV, pure infiniteness. In preparation.
- [Ka5]
Katsura, T. On
-algebras associated with
-correspondences. Preprint 2003, math.OA/0309088, to appear in J. Funct. Anal.
- [KPR] Alex Kumjian, David Pask, and Iain Raeburn, Cuntz-Krieger algebras of directed graphs, Pacific J. Math. 184 (1998), no. 1, 161–174. MR 1626528, https://doi.org/10.2140/pjm.1998.184.161
- [KPRR] Alex Kumjian, David Pask, Iain Raeburn, and Jean Renault, Graphs, groupoids, and Cuntz-Krieger algebras, J. Funct. Anal. 144 (1997), no. 2, 505–541. MR 1432596, https://doi.org/10.1006/jfan.1996.3001
- [L] E. C. Lance, Hilbert 𝐶*-modules, London Mathematical Society Lecture Note Series, vol. 210, Cambridge University Press, Cambridge, 1995. A toolkit for operator algebraists. MR 1325694
- [MS] Paul S. Muhly and Baruch Solel, Tensor algebras over 𝐶*-correspondences: representations, dilations, and 𝐶*-envelopes, J. Funct. Anal. 158 (1998), no. 2, 389–457. MR 1648483, https://doi.org/10.1006/jfan.1998.3294
- [P] Michael V. Pimsner, A class of 𝐶*-algebras generalizing both Cuntz-Krieger algebras and crossed products by 𝑍, Free probability theory (Waterloo, ON, 1995) Fields Inst. Commun., vol. 12, Amer. Math. Soc., Providence, RI, 1997, pp. 189–212. MR 1426840
- [RaSz] Raeburn, I.; Szymanski, W. Cuntz-Krieger algebras of infinite graphs and matrices. Preprint.
- [RoSc] Jonathan Rosenberg and Claude Schochet, The Künneth theorem and the universal coefficient theorem for Kasparov’s generalized 𝐾-functor, Duke Math. J. 55 (1987), no. 2, 431–474. MR 894590, https://doi.org/10.1215/S0012-7094-87-05524-4
- [Sc] Jürgen Schweizer, Crossed products by 𝐶*-correspondences and Cuntz-Pimsner algebras, 𝐶*-algebras (Münster, 1999) Springer, Berlin, 2000, pp. 203–226. MR 1798598
- [Sz] Wojciech Szymański, On semiprojectivity of 𝐶*-algebras of directed graphs, Proc. Amer. Math. Soc. 130 (2002), no. 5, 1391–1399. MR 1879962, https://doi.org/10.1090/S0002-9939-01-06282-7
- [T1] Jun Tomiyama, The interplay between topological dynamics and theory of 𝐶*-algebras, Lecture Notes Series, vol. 2, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1992. MR 1160781
- [T2] Jun Tomiyama, Structure of ideals and isomorphisms of 𝐶*-crossed products by single homeomorphism, Tokyo J. Math. 23 (2000), no. 1, 1–13. MR 1763501, https://doi.org/10.3836/tjm/1255958804
- [T3]
Tomiyama, J. Hulls and kernels from topological dynamical systems and their applications to homeomorphism
-algebras. To appear in J. Math. Soc. Japan.
- [T4]
Tomiyama, J. On the projection theorem for homeomorphism
-algebras. Preprint.
- [W] Simon Wassermann, Exact 𝐶*-algebras and related topics, Lecture Notes Series, vol. 19, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1994. MR 1271145
Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 46L05, 46L55, 37B99
Retrieve articles in all journals with MSC (2000): 46L05, 46L55, 37B99
Additional Information
Takeshi Katsura
Affiliation:
Department of Mathematical Sciences, University of Tokyo, Komaba, Tokyo 153-8914, Japan
Address at time of publication:
Department of Mathematics, Hokkaido University, Sapporo, Hokkaido 060-0810, Japan
Email:
katsura@math.sci.hokudai.ac.jp
DOI:
https://doi.org/10.1090/S0002-9947-04-03636-0
Received by editor(s):
October 1, 2002
Published electronically:
May 28, 2004
Article copyright:
© Copyright 2004
American Mathematical Society