The nuclear dimension of graphC⁎-algebras

Abstract

Consider a graph C * -algebra C * (E) with a purely infinite ideal I (possibly all of C * (E)) such that I has only finitely many ideals and C * (E)/I is approximately finite dimensional. We prove that the nuclear dimension of C * (E) is 1. If I has infinitely many ideals, then the nuclear dimension of C * (E) is either 1 or 2.

UCT-Kirchberg algebras have nuclear dimension one

“…For the duration of this section, we fix a row-finite k-graph Λ with no sources. The key tool for understanding nuclear dimension of graph algebras in [19] was a construction due to Kribs and Solel [8]. The first step in our analysis here is to adapt this construction to k-graphs.…”

“…For each n we now construct a homomorphism from C * (Λ) to C * (Λ(n)) analogous to those for directed graphs described in [19,Lemma 2.5].…”

“…We recall the definition of the Toeplitz algebra T C * (E) and the Cuntz-Krieger algebra C * (E) of a directed graph E as used in [19]. Let E = (E 0 , E 1 , r E , s E ) be a row-finite directed graph with no sinks (so 0 < | {e ∈ E 1 | s E (e) = v} | < ∞ for v ∈ E 0 ).…”

“…Here the underlying combinatorial model plays a key role. For graph C * -algebras [19], following [26,Section 7], the n th order-1 factorisation through finite-dimensional C * -algebras was obtained by producing a pair of pavings of the combinatorial quadrant N × N by copies of a completely positive contractive n × n real matrix so that when the two pavings are superimposed, entries at any fixed distance from the diagonal, and sufficiently far from the origin, approach 1 as n → ∞. For 2-graphs, we must perform an analogous decomposition in N 2 × N 2 , and the expected increase in dimension turns out to be illusory: facing no topological constraints, we can just pick convenient bijections of N 2 × N 2 onto N × N that carry points close to the diagonal to points close to the diagonal.…”

KMS States on Generalised Bunce–Deddens Algebras and their Toeplitz Extensions

“…Upon restriction to the canonical abelian subalgebra in T C * (E(mn)), these inclusions are compatible with a natural surjection E <mn → E <n , so lim − → T C * (E(n)) has an abelian subalgebra isomorphic to C 0 (lim ← − E <n ). This construction has recently been used to calculate the nuclear dimension of graph algebras and Kirchberg algebras [26,27].…”