A dual graph construction for higher-rank graphs, and K-theory for finite 2-graphs

Given a k-graph A and an element p of N k , we define the dual k-graph, pA. We show that when A is row-finite and has no sources, the C*-algebras C*(A) and C*(pA) coincide. We use this isomorphism to apply Robertson and Steger's results to calculate the K-theory of C*(A) when A is finite and strongly connected and satisfies the aperiodicity condition.

[1]  Guyan Robertson,et al.  Affine buildings, tiling systems and higher rank Cuntz-Krieger algebras , 2013, 1302.5593.

[2]  I. Raeburn,et al.  Cuntz-Krieger Algebras of Infinite Graphs and Matrices , 2003 .

[3]  T. Bates,et al.  Flow equivalence of graph algebras , 2002, Ergodic Theory and Dynamical Systems.

[4]  T. Bates.  Applications of the Gauge-invariant uniqueness theorem for graph algebras , 2002, Bulletin of the Australian Mathematical Society.

[5]  I. Raeburn,et al.  HIGHER-RANK GRAPHS AND THEIR $C^*$-ALGEBRAS , 2001, Proceedings of the Edinburgh Mathematical Society.

[6]  T. Steger,et al.  Asymptotic K-Theory for Groups Acting on à 2 Buildings , 2000, Canadian Journal of Mathematics.

[7]  E. Kirchberg,et al.  Embedding of exact C^*-algebras in the Cuntz algebra *_2 , 2000 .

[8]  I. Raeburn,et al.  The C^*-algebras of infinite graphs , 1999 .

[9]  I. Raeburn,et al.  CUNTZ-KRIEGER ALGEBRAS OF DIRECTED GRAPHS , 1998 .

[10]  J. Renault,et al.  Graphs, Groupoids, and Cuntz–Krieger Algebras , 1997 .

[11]  N. Phillips.  A classification theorem for nuclear purely infinite simple $C^*$-algebras , 1995, Documenta Mathematica.

[12]  C. Bennett.  Affine Λ‐Buildings, I , 1994 .

[13]  I. Raeburn,et al.  Representations of finite groups and Cuntz-Krieger algebras , 1992, Bulletin of the Australian Mathematical Society.

[14]  Wolfgang Krieger,et al.  A class ofC*-algebras and topological Markov chains , 1980 .

[15]  榎本 雅俊,et al.  A Graph Theory for $C^*$-Algebras (Operator Algebras and Their Applications ) , 1980 .

[16]  Wojciech,et al.  The range of K-invariants for C*-algebras of infinite graphs , 2002 .

[17]  A. Kumjian,et al.  On higher rank graph C ∗ -algebras , 2000 .

[18]  I. Raeburn,et al.  THE C -ALGEBRAS OF ROW-FINITE GRAPHS , 2000 .