Periodicity in Rank 2 Graph Algebras

Davidson, Kenneth R. ; Yang, Dilian

Canadian journal of mathematics, Tome 61 (2009) no. 6, pp. 1239-1261

Résumé

Kumjian and Pask introduced an aperiodicity condition for higher rank graphs. We present a detailed analysis of when this occurs in certain rank 2 graphs. When the algebra is aperiodic, we give another proof of the simplicity of ${{\text{C}}^{*}}\left( \mathbb{F}_{\theta }^{+} \right)$ . The periodic ${{\text{C}}^{*}}$ -algebras are characterized, and it is shown that ${{\text{C}}^{*}}\left( \mathbb{F}_{\theta }^{+} \right)\,\simeq \,\text{C}\left( \mathbb{T} \right)\,\otimes \,\mathfrak{U}$ where $\mathfrak{A}$ is a simple ${{\text{C}}^{*}}$ -algebra.

Détail

DOI : 10.4153/CJM-2009-058-0

Mots-clés : 47L55, 47L30, 47L75, 46L05, higher rank graph, aperiodicity condition, simple C*-algebra, expectation

Bibliographie

[1] [1] Cuntz, J., Simple C*-algebras generated by isometries. Comm. Math. Phys. 57(1977), no. 2, 173–185. Google Scholar

[2] [2] Davidson, K. R., C-Algebras by Example. Fields Institute Monograph Series 6, American Mathematical Society, Providence, RI, 1996. Google Scholar

[3] [3] Davidson, K. R., Power, S. C., and Yang, Dilian, Dilation theory for rank 2 graph algebras. To appear in J. Operator Theory. Google Scholar

[4] [4] Davidson, K. R., Power, S. C., Atomic representations of rank 2 graph algebras. J. Funct. Anal. 255(2008), 819–853. Google Scholar

[5] [5] Farthing, C., Muhly, P. S., and Yeend, T., Higher-rank graph C-algebras: an inverse semigroup and groupoid approach. Semigroup Forum 71(2005), no. 2, 159–187. Google Scholar

[6] [6] Kribs, D.W. and Power, S. C., The analytic algebras of higher rank graphs. Math. Proc. R. Ir. Acad. (2006), no. 2, 199–218. Google Scholar

[7] [7] Kumjian, A. and Pask, D., Higher rank graph C-algebras, New York J. Math. 6(2000), 1–20. Google Scholar

[8] [8] Pask, D., Raeburn, I., Rordam, M. and Sims, A., Rank-two graphs whose C*-algebras are direct limits of circle algebras. J. Funct. Anal. 239(2006), no. 1, 137–178. Google Scholar

[9] [9] Power, S.C., Classifying higher rank analytic Toeplitz algebras. New York J. Math. 13(2007), 271–298 (electronic). Google Scholar

[10] [10] Raeburn, I., Graph Algebras. CB MS Regional Converence Series in Mathematics 103. American Mathematical Society, Providence, RI, 2005. Google Scholar

[11] [11] Raeburn, I., Sims, A. and Yeend, T., Higher-rank graphs and their C-algebras. Proc. Edinb. Math. Soc. 46(2003), no. 1, 99–115. Google Scholar

[12] [12] Raeburn, I., Sims, A. and Yeend, T., The C*-algebras of finitely aligned higher-rank graphs. J. Funct. Anal. 213(2004), no. 1, 206–240. Google Scholar

[13] [13] Robertson, D. I. and Sims, A., Simplicity of C*-algebras associated to higher-rank graphs. Bull. Lond. Math. Soc. 39(2007), no. 2, 337–344. Google Scholar

[14] [14] Robertson, G. and Steger, T., Affine buildings, tiling systems and higher rank Cuntz-Krieger algebras. J. Reine Angew. Math. 513(1999), 115–144. Google Scholar

[15] [15] Sims, A., Gauge-invariant ideals in the C*-algebras of finitely aligned higher-rank graphs. Canad. J. Math. 58(2006), no. 6, 1268–1290. Google Scholar