AF-embeddability of $2$-graph algebras and quasidiagonality of $k$-graph algebras

TL;DR

This paper characterizes when the $C^*$-algebra of a cofinal $k$-graph is quasidiagonal using algebraic conditions on the graph's matrices, and shows equivalences among AF-embeddability, quasidiagonality, and stable finiteness for cofinal 2-graphs.

Contribution

It provides a complete algebraic characterization of quasidiagonality for cofinal $k$-graph $C^*$-algebras and establishes equivalences for cofinal 2-graphs.

Findings

Quasidiagonality characterized by algebraic conditions on coordinate matrices.

For cofinal 2-graphs, AF-embeddability, quasidiagonality, and stable finiteness are equivalent.

Results apply to all simple $k$-graph $C^*$-algebras.

Abstract

We characterise quasidiagonality of the $C^*$-algebra of a cofinal $k$-graph in terms of an algebraic condition involving the coordinate matrices of the graph. This result covers all simple $k$-graph $C^*$-algebras. In the special case of cofinal $2$-graphs we further prove that AF-embeddability, quasidiagonality and stable finiteness of the $2$-graph algebra are all equivalent.