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

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 ℓ2-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 lth higher shift associated to the one-sided shift space of E.