Viewing AF-algebras as graph algebras

Author:
D. Drinen

Journal:
Proc. Amer. Math. Soc. **128** (2000), 1991-2000

MSC (1991):
Primary 46L05; Secondary 22A22

DOI:
https://doi.org/10.1090/S0002-9939-99-05286-7

Published electronically:
November 1, 1999

MathSciNet review:
1657723

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Abstract | References | Similar Articles | Additional Information

Abstract: Every AF-algebra arises as the -algebra of a locally finite pointed directed graph in the sense of Kumjian, Pask, Raeburn, and Renault. For AF-algebras, the diagonal subalgebra defined by Stratila and Voiculescu is consistent with Kumjian's notion of diagonal, and the groupoid arising from a well-chosen Bratteli diagram for coincides with Kumjian's twist groupoid constructed from a diagonal of .

**1.**C. Anantharaman-Delaroche and J. Renault,*Amenable groupoids*, Mem. Amer. Math. Soc., (to appear).**2.**O. Bratteli,*Inductive limits of finite dimensional -algebras*, Trans. Amer. Math. Soc.**171**(1972), 195-234. MR**47:844****3.**E. Effros,*Dimensions and -algebras*, CBMS Regional Conf. Ser. in Math. no. 46, American Mathematical Society, Providence, RI, 1980. MR**84k:46042****4.**A. Kumjian,*On -diagonals*, Can. J. Math.**38**(1986), 969-1008. MR**88a:46060****5.**A. Kumjian, D. Pask, and I. Raeburn,*Cuntz-Krieger algebras of directed graphs*, Pacific J. Math.**184**(1998), 161-174. CMP**98:13****6.**A. Kumjian, D. Pask, I. Raeburn, and J. Renault,*Graphs, groupoids, and Cuntz-Krieger algebras*, J. Funct. Anal.**144**(1997), 505-541. MR**98g:46083****7.**S.C. Power,*Limit algebras*, Pitman Research Notes in Mathematics, vol. 278, Longman Scientific and Technical, London, 1992. MR**94g:46001****8.**J. N. Renault,*A groupoid approach to -algebras*, Lecture Notes in Math., no. 793, Springer-Verlag, New York, 1980. MR**92h:46075****9.**S. Str[??]atil[??]a and D. Voiculescu,*Representations of AF-algebras and of the group*, Lecture Notes in Math., no. 486, Springer, New York, 1975. MR**56:16391**

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Additional Information

**D. Drinen**

Affiliation:
Department of Mathematics, Arizona State University, Tempe, Arizona 85287

Email:
Drinen@asu.edu

DOI:
https://doi.org/10.1090/S0002-9939-99-05286-7

Received by editor(s):
March 23, 1998

Received by editor(s) in revised form:
August 18, 1998

Published electronically:
November 1, 1999

Communicated by:
David R. Larson

Article copyright:
© Copyright 2000
American Mathematical Society