Topological higher-rank graphs and the 𝐶*-algebras of topological 1-graphs

Abstract

We provide groupoid models for Toeplitz and Cuntz-Krieger algebras of topological higher-rank graphs. Extending the groupoid models used in the theory of graph algebras and topological dynamical systems to our setting, we prove results on essential freeness and amenability of the groupoids which capture the existing theory, and extend results involving group crossed products of graph algebras.Notation 2.2. For m ∈ N k , we write m i for the ith coordinate of m. We use the partial ordering ≤ on N k defined by m ≤ n ⇐⇒ m i ≤ n i for all i ∈ {1, . . . , k}, so least upper bounds and greatest lower bounds are given by (m ∨ n) i = max{m i , n i } and (m ∧ n) i = min{m i , n i }, respectively. For m ∈ N k , define Λ m to be the setq for the set of minimal common extensions of paths from U and V . For λ, µ ∈ Λ, we write Λ min (λ, µ) := {(α, β)|λα = µβ, d(λα) = d(λ) ∨ d(µ)} for the set of pairs which give minimal common extensions of λ and µ; that is, Λ min (λ, µ) = {(α, β)|λα = µβ ∈ {λ} ∨ {µ}}. Definition 2.3. A topological k-graph (Λ, d) is compactly aligned if for all p, q ∈ N k and compact U ⊂ Λ p and V ⊂ Λ q , the set U ∨ V is compact.

Cartan subalgebras and the UCT problem

“…Every UCT Kirchberg algebra has a Cartan subalgebra. That follows from [9, Thereom C] and [28,29], see also [21]. As every separable, nuclear C*-algebra is KK-equivalent to a Kirchberg algebra by [10, Theorem I], we therefore conclude that the UCT problem has a positive answer if and only if every Kirchberg algebra admits a Cartan subalgebra.…”

“…In combination with results by Katsura [9] and Yeend [28,29], Corollary 1.2 in particular implies that a Kirchberg algebra satisfies the UCT if and only if it admits a Cartan subalgebra, see Remark 3.3.…”

Tensor algebras of product systems and their C⁎-envelopes

“…In this section we discuss our results in the case of operator algebras arising from higher rank graphs and show by example that Theorem 4.1 is optimal. Our main result from Section 3 has consequences on operator algebras arising from topological higher rank graphs in the sense of Yeend [57,58] or those arising from dynamical systems as in [13], and this is discussed at the end of the section.…”

“…To describe this class, one needs to introduce the class of topological higher-rank graphs. We will not do this here but instead we direct the reader to the papers of Yeend [57,58] for the pertinent definitions and additional details. The following is an immediate corollary of Theorem 3.3 and [11,Theorem 5.20].…”

Traces on topological-graph algebras

“…Most of the work involved is in proving the converse. The key observation is that given a tracial state τ on C * (E), the GNS space L 2 (C * (E), τ ) decomposes as a direct integral over the boundary path space ∂E developed by Yeend in [17] and [18]. The fibres of the direct integral and the measure on the base space ∂E can be described explicitly in terms of the graph E and the invariant measure µ on E. Hence it is possible to build the GNS representation of the desired tracial state using only the invariant measure on the graph and then the trace is constructed as a vector state on this space.…”