Abstract
In this paper, we consider the relation between index theory and \(K\)-theory induced by directed graphs. In particular, we study index-morphism on finite trees, and classify the set of finite trees in terms of our index-morphism. Such a morphism generate certain semigroup, called the index semigroup. From the index semigroup, we find a ple, interesting connection between semigroup-elements and \(K\)-group computations of groupoid \(C^{*}\)-algebras generated by graphs. In conclusion, we show that the pure combinatorial data of graphs completely characterize and classify the elements of the index semigroup (or equivalently, graph-index on finite trees), Watatani’s Jones index on groupoid \(C^{*}\)-algebras generated by finite trees, and \(K\)-group computations of certain \(C^{*}\)-algebras.
Similar content being viewed by others
References
Cho, I.: Graph Groupoids and Partial Isometries. LAP Publisher. ISBN: 978-3-8383-1397-9 (2009)
Cho, I.: \(K\)-Theory on Groupoid Algebras Induced by Graphs (2011, preprint)
Cho, I.: Operations on Graphs, Groupoids, and Operator Algebras. LAP Publisher. ISBN: 978-8383-5271-8 (2010)
Cho, I.: Operator algebraic quotient structures induced by directed graphs. Comp. Anal. Oper. Theo. doi:10.1063/1.3056588 (2010)
Cho, I.: Index on von Neumann algebras induced by graphs. Appl. Math. Sci. (2012, to appear)
Cho, I.: Index-morphism on Finite trees and classification of von Neumann algebras induced by finite trees. Quantum Mech. InTech Publisher. ISBN: 979-953-307-377-3 (2011)
Cho, I., Jorgensen, P.E.T.: \(C^{*}\)-subalgebras generated by a single operator in \(B(H)\). J. Math. Phys. doi:10.1007s/10440-009-9478-5 (2008)
Cho, I., Jorgensen, P.E.T.: Directed graphs, von Neumann algebras, and index. Alg. Rep. Theo. doi:10.1007/s10468-010-9233-7 (2010)
Frank, M., Kirchberg, E.: On conditional expectations of finite index. J. Oper. Theo. 40, 87–111 (1998)
Jones, V.F.R.: Index for subfactors. Inv. Math. 72, 1–25 (1983)
Kosaki, H.: Extension of Jones’ theory on index to arbitrary factors. J. Funct. Anal. 66, 123–140 (1986)
Ocneanu, A.: Quantized Groups, String Algebras and Galois Theory for Algebras (1985, preprint)
Pimsner, M., Popa, S.: Entropy and index for subfactors. Ann. Sci. Ec. Norm. Sup. 19, 57–106 (1985)
Pimsner, M., Voiculescu, D.: \(K\)-Grouops of Reduced Crossed Products by Free Groups. J. Oper. Theo. 8, 131–156 (1982)
Speicher, R.: Combinatorial Theory of the Free Product with Amalgamation and Operator-Valued Free Probability Theory, vol. 132. Memoir Amar. Math. Soc., AMS (1998)
Voiculescu, D., Dykemma, K., Nica, A.: Free Random Variables, CRM Monograph Series, vol. 1. Amer. Math. Soc., (1992)
Watatani, Y.: Index for \(C^{*}\)-Subalgebras, vol. 424. Memoir, Am. Math. Soc., AMS (1990)
Du, Z., Zhou, B.: A note on Wiener indices of unicyclic graph. Ars. Combin. 93, 97–103 (2009)
Behtoei, A., Jannesari, M., Taeri, B.: Maximum Zagreb index, minimum hyper-Wiener index and graph connectivity. Appl. Math. Lett. 22(10), 1571–1576 (2009)
Dankelmann, P., Gutman, I., Mukwembi, S., Swart, H.C.: The edge-Wiener index of a graph. Discrete Math. 309(10), 3452–3457 (2009)
Iranmanesh, A., Gutman, I., Khormali, O., Mahmini, A.: The edge version of the Wiener index. MATCH Commun. Math. Comput. Chem. 61(3), 663–672 (2009)
Geng, X., Li, S., Li, X.: On the index of tricyclic graphs with perfct matchings. Linear Algebra Appl. 431(12), 2304–2316 (2009)
Mitchener, P.D.: \(C^{*}\)-Categories, Groupoid Actions, Equivalent \(KK\)-Theory, and the Baum-Connes Conjecture. arXiv:math.KT/0204291v1 (2005, preprint)
Dicks, W., Ventura, E.: The group fixed by a family of injective endomorphisms of a free group. Contemp. Math 195. ISBN: 0-8218-0564-9
Davidson, K.R.: \(C^{*}\)-Algebras by Example, Field Institute Monographs, vol. 6. Amer. Math. Soc., Providence (1996)
Brodzki, J.: An Introduction to \(K\)-Theory and Cyclic Cohomology, Lecture Note, Department of Mathematics, U. of Exeter (1995)
Atiyah, M.: \(K\)-Theory. Benjamin, New York (1967)
Connes, A.: Noncommutative differential geometry. Publ. Math. IHES 62, 257–360 (1985)
Cuntz, J.: \(K\)-Theory and \(C^{*}\)-Algebras, vol. 1046. Lecture Notes in Math. Springer, Berlin, pp. 55–79
Cuntz, J.: A new look at \(KK\)-Theory. \(K\)-Theory 1, 31–51 (1987)
Goodwillie, T.: Cyclic cohomology, derivations and the free loop space. Topology 24, 187–215 (1985)
Higson, N.: A Primer on \(KK\)-Theory. Proc. Symp. Pure Math. 51, 239–284
Jaffe, A., Lesniewski, A., Osterwalder, K.: Quantum \(K\)-theory: the Chern character. Commun. Math. Phys. 118, 1–14 (1988)
Milnor, J.: Algebraic \(K\)-Theory, Annals of Mathematics Studies, vol. 72. Princeton University Press, Princeton (1971)
Rosenberg, J.: \(K\) and \(KK\): topology and operator algebras. Proc. Symp. Pure Math. 51(Part 1), 445–480 (1990)
Rosenberg, J.: Algebraic \(K\)-Theory and its Applications. Graduate Text in Mathematics, vol. 147. Springer, Berlin (1994)
Swan, R.: Excision in algebraic \(K\)-theory. J. Pure Appl. Algebra 1, 221–252 (1972)
Suslin, A., Wodzicki, M.: Excision in algebraic \(K\)-theory. Ann. Math. 136, 51–122 (1992)
Kriz, I., Sati, H.: Type \(IIB\) string theory, \(S\)-duality, and generalized cohomology. Nucl. Phys. B 715, 639–664 (2005)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Palle Jorgensen.
Rights and permissions
About this article
Cite this article
Cho, I. The Index Semigroup and \(K\)-Groups of Graph-Groupoid \(C^{*}\)-Algebras. Complex Anal. Oper. Theory 8, 57–109 (2014). https://doi.org/10.1007/s11785-012-0271-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11785-012-0271-5
Keywords
- Finite Directed Graphs
- Graph Groupoids
- Graph von Neumann Algebras
- Graph Inclusions
- Graph-Index
- Jones Index
- Quasi-bases
- Watatani’s Extended Jones Index
- Index-Morphism
- Index Semigroups