Endomorphisms of graph algebras

We initiate a systematic investigation of endomorphisms of graph C⁎C⁎-algebras C⁎(E)C⁎(E), extending several known results on endomorphisms of the Cuntz algebras OnOn. Most but not all of this study is focused on endomorphisms which permute the vertex projections and globally preserve the diagonal MASA DEDE of C⁎(E)C⁎(E). Our results pertain both automorphisms and proper endomorphisms. Firstly, the Weyl group and the restricted Weyl group of a graph C⁎C⁎-algebra are introduced and investigated. In particular, criteria of outerness for automorphisms in the restricted Weyl group are found. We also show that the restriction to the diagonal MASA of an automorphism which globally preserves both DEDE and the core AF-subalgebra eventually commutes with the corresponding one-sided shift. Secondly, we exhibit several properties of proper endomorphisms, investigate invertibility of localized endomorphisms both on C⁎(E)C⁎(E) and in restriction to DEDE, and develop a combinatorial approach to analysis of permutative endomorphisms.