Localized Endomorphisms of Graph Algebras

Abstract

Endomorphisms of graph $C^*$-algebras are investigated. A combinatorial approach to analysis of permutative endomorphisms is developed. Then invertibility criteria for localized endomorphisms are given. Furthermore, proper endomorphisms which restrict to automorphisms of the canonical diagonal MASA are analyzed. The Weyl group and the restricted Weyl group of a graph $C^*$-algebra are introduced and investigated. Criteria of outerness for automorphisms in the restricted Weyl group are found.