## Annals of Functional Analysis

### On certain properties of Cuntz–Krieger-type algebras

#### Abstract

This note presents a further study of the class of Cuntz–Krieger-type algebras. A necessary and sufficient condition is identified that ensures that the algebra is purely infinite, the ideal structure is studied, and nuclearity is proved by presenting the algebra as a crossed product of an AF-algebra by an abelian group. The results are applied to examples of Cuntz–Krieger-type algebras, such as higher-rank semigraph $C^{*}$-algebras and higher-rank Exel–Laca algebras.

#### Article information

Source
Ann. Funct. Anal., Volume 8, Number 3 (2017), 386-397.

Dates
Accepted: 6 November 2016
First available in Project Euclid: 9 May 2017

https://projecteuclid.org/euclid.afa/1494295270

Digital Object Identifier
doi:10.1215/20088752-2017-0004

Mathematical Reviews number (MathSciNet)
MR3690001

Zentralblatt MATH identifier
1380.46038

#### Citation

Burgstaller, Bernhard; Evans, D. Gwion. On certain properties of Cuntz–Krieger-type algebras. Ann. Funct. Anal. 8 (2017), no. 3, 386--397. doi:10.1215/20088752-2017-0004. https://projecteuclid.org/euclid.afa/1494295270

#### References

• [1] O. Bratteli, Inductive limits of finite dimensional $C^{*}$-algebras, Trans. Amer. Math. Soc. 171 (1972), 195–234.
• [2] B. Burgstaller, The uniqueness of Cuntz–Krieger type algebras, J. Reine Angew. Math. 594 (2006), 207–236.
• [3] B. Burgstaller, A class of higher rank Exel–Laca algebras, Acta Sci. Math. (Szeged) 73 (2007), no. 1–2, 209–235.
• [4] B. Burgstaller, A Cuntz–Krieger uniqueness theorem for semigraph $C^{*}$-algebras, Banach J. Math. Anal. 6 (2012), no. 2, 38–57.
• [5] B. Burgstaller, Representations of crossed products by cancelling actions and applications, Houston J. Math. 38 (2012), no. 3, 761–774.
• [6] J. Cuntz, Simple $C^{*}$-algebras generated by isometries, Commun. Math. Phys. 57 (1977), 173–185.
• [7] J. Cuntz, A class of $C^{*}$-algebras and topological Markov chains, II: Reducible chains and the Ext-functor for $C^{*}$-algebras, Invent. Math. 63 (1981), no. 1, 25–40.
• [8] J. Cuntz and W. Krieger, A class of $C^{*}$-algebras and topological Markov chains, Invent. Math. 56 (1980), 251–268.
• [9] S. Eilers, T. Katsura, M. Tomforde, and J. West, The ranges of K-theoretic invariants for nonsimple graph algebras, Trans. Amer. Math. Soc. 368 (2016), no. 6, 3811–3847.
• [10] E. Gillaspy, K-theory and homotopies of 2-cocycles on higher-rank graphs, Pacific J. Math. 278 (2015), no. 2, 407–426.
• [11] A. Kumjian and D. Pask, Higher rank graph $C^{*}$-algebras, New York J. Math. 6 (2000), 1–20.
• [12] I. Raeburn, A. Sims, and T. Yeend, The $C^{*}$-algebras of finitely aligned higher-rank graphs, J. Funct. Anal. 213 (2004), no. 1, 206–240.
• [13] I. Raeburn and W. Szymański, Cuntz–Krieger algebras of infinite graphs and matrices, Trans. Amer. Math. Soc. 356 (2004), no. 1, 39–59.
• [14] G. Robertson and T. Steger, Affine buildings, tiling systems and higher rank Cuntz–Krieger algebras, J. Reine Angew. Math. 513 (1999) 115–144.
• [15] M. Rørdam and E. Størmer, Classification of Nuclear $C^{*}$-algebras: Entropy in Operator Algebras, Encyclopaedia Math. Sci. 126, Springer, Berlin, 2002.
• [16] H. Takai, On a duality for crossed products of $C^{*}$-algebras, J. Funct. Anal. 19 (1975), 25–39.
• [17] M. Tomforde, A unified approach to Exel–Laca algebras and $C^{*}$-algebras associated to graphs, J. Operator Theory 50 (2003), no. 2, 345–368.