Co-universal C*-algebras associated to aperiodic k-graphs

TL;DR

This paper constructs a specific representation of aperiodic k-graphs and demonstrates its co-universality property for Toeplitz-Cuntz-Krieger families, advancing the understanding of their associated C*-algebras.

Contribution

It introduces a new representation of aperiodic k-graph C*-algebras that is co-universal for certain families, providing a novel structural insight.

Findings

The constructed representation is isomorphic to the Cuntz-Krieger algebra.

The expectation on the representation restricts to a subalgebra spanned by final projections.

Every quotient of the Toeplitz algebra admits a compatible expectation.

Abstract

We construct a representation of each finitely aligned aperiodic k-graph \Lambda\ on the Hilbert space H^{ap} with basis indexed by aperiodic boundary paths in \Lambda. We show that the canonical expectation on B(H^{ap}) restricts to an expectation of the image of this representation onto the subalgebra spanned by the final projections of the generating partial isometries. We then show that every quotient of the Toeplitz algebra of the k-graph admits an expectation compatible with this one. Using this, we prove that the image of our representation, which is canonically isomorphic to the Cuntz-Krieger algebra, is co-universal for Toeplitz-Cuntz-Krieger families consisting of nonzero partial isometries.