Endomorphisms of graph algebras

TL;DR

This paper systematically studies endomorphisms of graph C*-algebras, extending known results from Cuntz algebras, focusing on automorphisms and proper endomorphisms that permute vertices and preserve the diagonal MASA.

Contribution

It introduces the Weyl group concepts for graph C*-algebras, provides criteria for outerness of automorphisms, and develops a combinatorial approach to permutative endomorphisms.

Findings

Criteria for outerness of automorphisms in the restricted Weyl group

Properties and invertibility of proper endomorphisms

A combinatorial method for analyzing permutative endomorphisms

Abstract

We initiate a systematic investigation of endomorphisms of graph C*-algebras C*(E), extending several known results on endomorphisms of the Cuntz algebras O_n. Most but not all of this study is focused on endomorphisms which permute the vertex projections and globally preserve the diagonal MASA D_E of C*(E). Our results pertain both automorphisms and proper endomorphisms. Firstly, the Weyl group and the restricted Weyl group of a graph 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 the diagonal 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) and in restriction to D_E, and develop a combinatorial approach to analysis of permutative endomorphisms.