C*-equivalences of k-graph and N-graph algebras through graph transformations

Abstract

In the study of operator algebras, C*-algebras act as a generalization of matrix algebras over a vector space. A rich source of C*-algebras to study is the graph algebra, where functions are chosen based on the vertices and edges of a directed graph. Eilers and Ruiz proved in 2019 that many transformations of the graph-insplits, outsplits, and others-do not affect the ideal structure of the graph algebra. In 2020, several of these transformations of k-graphs (a higher-rank analog of directed graphs) were also shown to preserve Morita equivalence but the outsplit is missing from this list. We expand on this previous work by showing that the outsplit of a higher-rank graph will preserve Morita equivalence as well. We then begin to elevate this discussion to N-graph algebras by showing that sink deletion, delay, and reduction also preserve Morita equivalence.