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

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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**

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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