A New Geometric Morita Invariant for Higher Rank Graph $C^{\ast }$-algebras

Abstract

Higher rank graphs, also known as $k$-graphs, are a $k$-dimensional generalization of directed graphs and a rich source of examples of $C^{\ast }$-algebras. In the present paper, we contribute to the geometric classification program for $k$-graph $C^{\ast }$-algebras by introducing a new move on $k$-graphs, called LiMaR-split, which is a generalization of outsplit for directed graphs. We show, under one additional assumption, that LiMaR-split preserves the $k$-graph $C^{\ast }$-algebras up to Morita equivalence.

25 pages; changed title, added author, removed the pairing condition from Section 4 (revised and added lemmas), removed $C^{\ast }$-algebra lemmas from Section 4, strengthened Corollary 4.11 and added Corollary 4.12