A class of algebras generalizing both graph algebras and homeomorphism algebras I, fundamental results
Author:
Takeshi Katsura
Journal:
Trans. Amer. Math. Soc. 356 (2004), 42874322
MSC (2000):
Primary 46L05; Secondary 46L55, 37B99
Published electronically:
May 28, 2004
MathSciNet review:
2067120
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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 socalled gaugeinvariant uniqueness theorem and the CuntzKrieger uniqueness theorem, and compute the groups of our algebras.
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 Archbold, R. J.; Spielberg, J. S. Topologically free actions and ideals in discrete dynamical systems. Proc. Edinburgh Math. Soc. (2) 37 (1994), no. 1, 119124. MR 94m:46101
 [AR]
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 [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, 11591176.
 [BPRS]
 Bates, T.; Pask, D.; Raeburn, I.; Szymanski, W. The algebras of rowfinite graphs. New York J. Math. 6 (2000), 307324. MR 2001k:46084
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 Cuntz, J.; Krieger, W. A class of algebras and topological Markov chains. Invent. Math. 56 (1980), no. 3, 251268. MR 82f:46073a
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 Deaconu, V. Continuous graphs and C*algebras. Operator theoretical methods, 137149, Theta Found., Bucharest, 2000. MR 2001g:46123
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 Deaconu, V.; Kumjian, A.; Muhly, P. Cohomology of topological graphs and CuntzPimsner algebras. J. Operator Theory 46 (2001), no. 2, 251264. MR 2003a:46093
 [DT1]
 Drinen, D.; Tomforde, M. The algebras of arbitrary graphs. To appear in Rocky Mountain J. Math.
 [DT2]
 Drinen, D.; Tomforde, M. Computing theory and for graph algebras. Illinois J. Math. 46 (2002), no. 1, 8191. MR 2003k:46103
 [DS]
 Dykema, K.; Shlyakhtenko, D. Exactness of CuntzPimsner C*algebras. Proc. Edinburgh Math. Soc. 44 (2001), 425444. MR 2003a:46084
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 Exel, R. A Fredholm operator approach to Morita equivalence. Theory 7 (1993), no. 3, 285308. MR 94h:46107
 [EL]
 Exel, R.; Laca, M. CuntzKrieger algebras for infinite matrices. J. Reine Angew. Math. 512 (1999), 119172. MR 2000i:46064
 [ELQ]
 Exel, R.; Laca, M.; Quigg, J. Partial dynamical systems and algebras generated by partial isometries. J. Operator Theory 47 (2002), no. 1, 169186. MR 2003f:46108
 [FLR]
 Fowler, N. J.; Laca, M.; Raeburn, I. The algebras of infinite graphs. Proc. Amer. Math. Soc. 128 (2000), no. 8, 23192327. MR 2000k:46079
 [FR]
 Fowler, N. J.; Raeburn, I. The Toeplitz algebra of a Hilbert bimodule. Indiana Univ. Math. J. 48 (1999), no. 1, 155181. MR 2001b:46093
 [HS]
 Hong, J. H.; Szymanski, W. The primitive ideal space of the algebras of infinite graphs. J. Math. Soc. Japan 56 (2004), no. 1, 4564.
 [KPW]
 Kajiwara, T.; Pinzari, C.; Watatani, Y. Ideal structure and simplicity of the algebras generated by Hilbert bimodules. J. Funct. Anal. 159 (1998), no. 2, 295322. MR 2000a:46094
 [Ka1]
 Katsura, T. The ideal structures of crossed products of Cuntz algebras by quasifree actions of abelian groups. Canad. J. Math. 55 (2003), no. 6, 13021338.
 [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]
 Kumjian, A.; Pask, D.; Raeburn, I. CuntzKrieger algebras of directed graphs. Pacific J. Math. 184 (1998), no. 1, 161174. MR 99i:46049
 [KPRR]
 Kumjian, A.; Pask, D.; Raeburn, I.; Renault, J. Graphs, groupoids, and CuntzKrieger algebras. J. Funct. Anal. 144 (1997), no. 2, 505541. MR 98g:46083
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 Lance, E. C. Hilbert modules. A toolkit for operator algebraists. London Mathematical Society Lecture Note Series, 210. Cambridge University Press, Cambridge, 1995. MR 96k:46100
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 Muhly, P. S.; Solel, B. Tensor algebras over correspondences: representations, dilations, and envelopes. J. Funct. Anal. 158 (1998), no. 2, 389457. MR 99j:46066
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 Pimsner, M. V. A class of algebras generalizing both CuntzKrieger algebras and crossed products by . Free probability theory, 189212, Fields Inst. Commun., 12, Amer. Math. Soc., Providence, RI, 1997. MR 97k:46069
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 Raeburn, I.; Szymanski, W. CuntzKrieger algebras of infinite graphs and matrices. Preprint.
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 Rosenberg, J.; Schochet, C. The Kunneth theorem and the universal coefficient theorem for Kasparov's generalized functor. Duke Math. J. 55 (1987), no. 2, 431474. MR 88i:46091
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 Schweizer, J. Crossed products by correspondences and CuntzPimsner algebras. algebras, 203226, Springer, Berlin, 2000. MR 2002f:46133
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 Szymanski, W. On semiprojectivity of algebras of directed graphs. Proc. Amer. Math. Soc. 130 (2002), no. 5, 13911399. MR 2003a:46083
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 Tomiyama, J. The interplay between topological dynamics and theory of algebras. Lecture Notes Series, 2. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1992. MR 93h:46097
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 Tomiyama, J. Structure of ideals and isomorphisms of crossed products by single homeomorphism. Tokyo J. Math. 23 (2000), no. 1, 113. MR 2001e:46117
 [T3]
 Tomiyama, J. Hulls and kernels from topological dynamical systems and their applications to homeomorphism algebras. To appear in J. Math. Soc. Japan.
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 Tomiyama, J. On the projection theorem for homeomorphism algebras. Preprint.
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Additional Information
Takeshi Katsura
Affiliation:
Department of Mathematical Sciences, University of Tokyo, Komaba, Tokyo 1538914, Japan
Address at time of publication:
Department of Mathematics, Hokkaido University, Sapporo, Hokkaido 0600810, Japan
Email:
katsura@math.sci.hokudai.ac.jp
DOI:
http://dx.doi.org/10.1090/S0002994704036360
PII:
S 00029947(04)036360
Received by editor(s):
October 1, 2002
Published electronically:
May 28, 2004
Article copyright:
© Copyright 2004
American Mathematical Society
