Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-27T06:41:58.608Z Has data issue: false hasContentIssue false
  • English
  • Français

Towards a K-theoretic characterization of graded isomorphisms between Leavitt path algebras

Published online by Cambridge University Press:  30 June 2014

P. Ara  and
E. Pardo
P. Ara*
Affiliation:
Departament de Matemàtiques, Facultat de Ciències, Edifici C, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain
E. Pardo*
Affiliation:
Departamento de Matemáticas, Facultad de Ciencias, Universidad de Cádiz, Campus de Puerto Real, 11510 Puerto Real (Cádiz), Spain
*
[email protected]
[email protected]
Get access

Abstract

In Hazrat gave a K-theoretic invariant for Leavitt path algebras as graded algebras. Hazrat conjectured that this invariant classifies Leavitt path algebras up to graded isomorphism, and proved the conjecture in some cases. In this paper, we prove that a weak version of the conjecture holds for all finite essential graphs.

Keywords

Leavitt path algebragraph C*-algebragraded algebraK-TheoryPrimary 16D70Secondary 46L55
Type
Research Article
Information
Journal of K-Theory , Volume 14 , Issue 2 , October 2014 , pp. 203 - 245
Copyright
Copyright © ISOPP 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

1.Abrams, G., Pino, G. Aranda, The Leavitt path algebra of a graph, J. Algebra 293 (2005), 319–334.CrossRefGoogle Scholar
2.Abrams, G., Pino, G. Aranda, Purely infinite simple Leavitt path algebras, J. Pure Appl. Algebra, 207 (2006), 553–563.CrossRefGoogle Scholar
3.Abrams, G., Ánh, P.N., Louly, A., Pardo, E., Flow invariants in the classification of Leavitt algebras, J. Algebra 333 (2011), 202–231.CrossRefGoogle Scholar
4.Ara, P., Brustenga, M., Cortiñas, G., K-theory of Leavitt path algebras, Münster J. Math. 2 (2009), 5–33.Google Scholar
5.Ara, P., Cortiñas, G., Tensor products of Leavitt path algebras, Proc. Amer. Math. Soc. 141 (2013), 2629–2639.CrossRefGoogle Scholar
6.Ara, P., González-Barroso, M.A., Goodearl, K.R., Pardo, E., Fractional skew monoid rings. J. Algebra 278(1) (2004), 104–126.Google Scholar
7.Ara, P., Moreno, M. A., Pardo, E., Nonstable K-Theory for graph algebras, Algebra Rep. Th. 10 (2007), 157–178.CrossRefGoogle Scholar
8.Ara, P., Pardo, E., Stable rank of Leavitt path algebras, Proc. Amer. Math. Soc. 136 (2008), 2375–2386.CrossRefGoogle Scholar
9.Pino, G. Aranda, Pardo, E., Molina, M. Siles, Exchange Leavitt path algebras and stable rank, J. Algebra 305 (2006), 912–936.CrossRefGoogle Scholar
10.Pino, G. Aranda, Perera, F., Molina, M. Siles, Eds., Graph algebras: bridging the gap between analysis and algebra, ISBN: 978-84-9747-177-0, University of Málaga Press, Málaga, Spain (2007).Google Scholar
11.Bates, T., Applications of the gauge-invariant uniqueness theorem for the Cuntz-Krieger algebras of directed graphs, Bull. Austral. Math. Soc. 65 (2002), 57–67.CrossRefGoogle Scholar
12.Bavula, V. V., The group of automorphisms of the algebra of polynomial integro-differential operators, J. Algebra 348 (2011), 233–263.Google Scholar
13.Franks, J., Flow equivalence of subshifts of finite type, Ergod. Th. & Dynam. Sys. 4 (1984), 53–66.Google Scholar
14.Goodearl, K. R. “Von Neumann Regular Rings”, Second Edition, Krieger, Malabar 1991.Google Scholar
15.Gordon, R., Green, E.L., Graded Artin algebras. J. Algebra 76 (1982), 111–137.CrossRefGoogle Scholar
16.Hazrat, R., The graded Grothendieck group and the classification of Leavitt path algebras, Math. Annalen 355 (2013), 273–325.Google Scholar
17.Hazrat, R., The dynamics of Leavitt path algebras, J. Algebra 384 (2013), 242–266.Google Scholar
18.Hazrat, R., Graded Rings and graded Grothendieck groups. Monograph, http:rhazrat.wordpress.com/preprint.Google Scholar
19.Kirchberg, E., The classification of purely infinite C*-algebras using Kasparov theory, preprint.Google Scholar
20.Leavitt, W.G., The module type of a ring, Trans. Amer. Math. Soc. 103 (1962), 113–130.Google Scholar
21.Lind, D., Marcus, B., “An Introduction to Symbolic Dynamics and Coding”, Cambridge Univ. Press, Cambridge, 1995, Reprinted 1999 (with corrections). ISBN 0-521-55900-6.CrossRefGoogle Scholar
22.Matsumoto, K., Classification of Cuntz-Krieger algebras by orbit equivalence of topological Markov shifts, Proc. Amer. Math. Soc. 141 (2013), 2329–2342.CrossRefGoogle Scholar
23.Matsumoto, K., K-groups of the full group actions on one-sided topological Markov shifts, Discrete Contin. Dyn. Syst. 33 (2013), 3753–3765.Google Scholar
24.Nǎstǎsescu, C., Oystaeyen, F. van, “Methods of graded rings”, Lecture Notes in Mathematics 1836, Springer Verlag, Berlin, 2004.Google Scholar
25.Paschke, W.L., The crossed product of a C*-algebra by an endomorphism, Proc. Amer. Math. Soc. 80 (1980), 113–118.Google Scholar
26.Phillips, N.C., A classification theorem for nuclear purely infinite simple C*-algebras, Doc. Math. 5 (2000), 49–114.CrossRefGoogle Scholar
27.Raeburn, I., “Graph Algebras”, CBMS Reg. Conf. Ser. Math. 103, Amer. Math. Soc., Providence, RI, 2005.Google Scholar
28.Rørdam, M., Classification of Cuntz-Krieger algebras, K-Theory 9 (1995), 31–58.CrossRefGoogle Scholar
29.Smith, S. Paul, Category equivalences involving graded modules over path algebras of quivers, Adv. Math. 230 (2012), 1780–1810.CrossRefGoogle Scholar
30.Tomforde, M., Uniqueness theorems and ideal structure for Leavitt path algebras, J. Algebra 318(1) (2007), 270–299.Google Scholar
31.Williams, R.F., Classification of subshifts of finite type, Ann. Math. 98(2) (1973), 120–153.Google Scholar