Visualizing automorphisms of graph algebras

Abstract

Graph C*-algebras have been celebrated as C*-algebras that can be seen, because many important properties may be determined by looking at the underlying graph. This paper introduces the permutation graph for a permutative endomorphism of a graph C*-algebra as a labeled directed multigraph that gives a visual representation of the endomorphism and facilitates computations. Combinatorial criteria have previously been developed for deciding when such an endomorphism is an automorphism, but here the question is reformulated in terms of the permutation graph and new proofs are given. Furthermore, it is shown how to use per- mutation graphs to efficiently generate exhaustive collections of permutative automorphisms. Permutation graphs provide a natural link to the textile systems representing induced endo- morphisms on the edge shift of the given graph, and this allows the powerful tools of the theory of textile systems developed by Nasu to be applied to the study of permutative endomorphisms.