Co-universal C*-algebras associated to generalised graphs

TL;DR

This paper introduces P-graphs, a generalization of directed graphs with semigroup degrees, and constructs co-universal C*-algebras that encompass a broad class of Kirchberg algebras, extending graph algebra theory.

Contribution

It defines P-graphs based on semigroup degrees, constructs their co-universal C*-algebras, and characterizes injectivity conditions, broadening the framework of graph C*-algebras.

Findings

Every finitely aligned P-graph admits a co-universal C*-algebra.

The co-universal C*-algebra is characterized by a coaction of G.

Every Kirchberg algebra is Morita equivalent to a C*-algebra of some (N^2 * N)-graph.

Abstract

We introduce P-graphs, which are generalisations of directed graphs in which paths have a degree in a semigroup P rather than a length in N. We focus on semigroups P arising as part of a quasi-lattice ordered group (G,P) in the sense of Nica, and on P-graphs which are finitely aligned in the sense of Raeburn and Sims. We show that each finitely aligned P-graph admits a C*-algebra C*_{min}(Lambda) which is co-universal for partial-isometric representations of Lambda which admit a coaction of G compatible with the P-valued length function. We also characterise when a homomorphism induced by the co-universal property is injective. Our results combined with those of Spielberg show that every Kirchberg algebra is Morita equivalent C*_{min}(Lambda) for some (N^2 * N)-graph Lambda.