Stability of -algebras associated to graphs

Author:
Mark Tomforde

Translated by:

Journal:
Proc. Amer. Math. Soc. **132** (2004), 1787-1795

MSC (2000):
Primary 46L55

Published electronically:
January 30, 2004

MathSciNet review:
2051143

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Abstract | References | Similar Articles | Additional Information

Abstract: We characterize stability of graph -algebras by giving five conditions equivalent to their stability. We also show that if is a graph with no sources, then is stable if and only if each vertex in can be reached by an infinite number of vertices. We use this characterization to realize the stabilization of a graph -algebra. Specifically, if is a graph and is the graph formed by adding a head to each vertex of , then is the stabilization of ; that is, .

**1.**T. Bates, J. H. Hong, I. Raeburn, and W. Szymanski,*The ideal structure of the -algebras of infinite graphs*, Illinois J. Math.**46**(2002), 1159-1176.**2.**Teresa Bates, David Pask, Iain Raeburn, and Wojciech Szymański,*The 𝐶*-algebras of row-finite graphs*, New York J. Math.**6**(2000), 307–324 (electronic). MR**1777234****3.**Bruce E. Blackadar,*Traces on simple AF 𝐶*-algebras*, J. Funct. Anal.**38**(1980), no. 2, 156–168. MR**587906**, 10.1016/0022-1236(80)90062-2**4.**Joachim Cuntz,*Dimension functions on simple 𝐶*-algebras*, Math. Ann.**233**(1978), no. 2, 145–153. MR**0467332****5.**Joachim Cuntz and Wolfgang Krieger,*A class of 𝐶*-algebras and topological Markov chains*, Invent. Math.**56**(1980), no. 3, 251–268. MR**561974**, 10.1007/BF01390048**6.**D. Drinen and M. Tomforde,*The -algebras of arbitrary graphs*, preprint (2000).**7.**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**, 10.1090/S0002-9939-99-05378-2**8.**Jacob v. B. Hjelmborg,*Purely infinite and stable 𝐶*-algebras of graphs and dynamical systems*, Ergodic Theory Dynam. Systems**21**(2001), no. 6, 1789–1808. MR**1869070**, 10.1017/S0143385701001857**9.**Jacob v. B. Hjelmborg and Mikael Rørdam,*On stability of 𝐶*-algebras*, J. Funct. Anal.**155**(1998), no. 1, 153–170. MR**1623142**, 10.1006/jfan.1997.3221**10.**Alex Kumjian, David Pask, and Iain Raeburn,*Cuntz-Krieger algebras of directed graphs*, Pacific J. Math.**184**(1998), no. 1, 161–174. MR**1626528**, 10.2140/pjm.1998.184.161**11.**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**, 10.1006/jfan.1996.3001**12.**Mark Tomforde,*The ordered 𝐾₀-group of a graph 𝐶*-algebra*, C. R. Math. Acad. Sci. Soc. R. Can.**25**(2003), no. 1, 19–25 (English, with English and French summaries). MR**1962131****13.**Yasuo Watatani,*Graph theory for 𝐶*-algebras*, Operator algebras and applications, Part I (Kingston, Ont., 1980) Proc. Sympos. Pure Math., vol. 38, Amer. Math. Soc., Providence, R.I., 1982, pp. 195–197. MR**679705**

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Additional Information

**Mark Tomforde**

Affiliation:
Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755-3551

Address at time of publication:
Department of Mathematics, University of Iowa, Iowa City, Iowa 52242

Email:
tomforde@math.uiowa.edu

DOI:
https://doi.org/10.1090/S0002-9939-04-07411-8

Received by editor(s):
June 14, 2002

Received by editor(s) in revised form:
March 1, 2003

Published electronically:
January 30, 2004

Communicated by:
David R. Larson

Article copyright:
© Copyright 2004
American Mathematical Society