The suspension of a graph, and associated C*-algebras

TL;DR

This paper introduces a new construction called the suspension of a graph, linking it to shift spaces and associated C*-algebras, revealing new algebraic and topological properties for different parameters.

Contribution

It constructs a novel class of quivers called suspensions of graphs and analyzes their associated C*-algebras, connecting them to shift spaces and higher shifts.

Findings

The suspension quiver's infinite-path space is homeomorphic to the suspension of the one-sided shift.

The Cuntz-Krieger algebra of the suspension admits a faithful representation on the -space of the shift suspension.

For certain graphs and rational l, the associated algebras are homotopy equivalent to those of higher shift graphs.

Abstract

Given a directed graph E, we construct for each real number l a quiver whose vertex space is the topological realisation of E, and whose edges are directed paths of length l in the vertex space. These quivers are not topological graphs in the sense of Katsura, nor topological quivers in the sense of Muhly and Tomforde. We prove that when l = 1 and E is finite, the infinite-path space of the associated quiver is homeomorphic to the suspension of the one-sided shift of E. We call this quiver the suspension of E. We associate both a Toeplitz algebra and a Cuntz-Krieger algebra to each of the quivers we have constructed, and show that when l = 1 the Cuntz-Krieger algebra admits a natural faithful representation on the \ell^2-space of the suspension of the one-sided shift of E. For graphs E in which sufficiently many vertices both emit and receive at least two edges, and for rational values of l, we show that the Toeplitz algebra and the Cuntz-Krieger algebra of the associated quiver are homotopy equivalent to the Toeplitz algebra and Cuntz-Krieger algebra respectively of a graph that can be regarded as encoding the l-th higher shift associated to the one-sided shift space of E.