Simplicity of algebras associated to étale groupoids

Uniqueness theorems for topological higher-rank graph 𝐶*-algebras

“…As in the proof of Theorem 5.1, G Λ is second-countable, locally compact, Hausdorff, étale and amenable. Hence Theorem 5.1 of [3] implies that C * (Λ) is simple if and only if G Λ is topologically principal and minimal. Theorem 5.1 of [27] implies that G Λ is topologically principal if and only if Λ satisfies condition (1) of Theorem 5.1.…”

“…Since this condition should be easier to check, we describe what it says for a topological k-graph: it is a topological analogue of the condition called "no local periodicity" in [20]. The third condition below is Wright's finite-paths aperiodicity condition [23, Theorem 3.1(C)]; as mentioned above, our argument below recovers the equivalence (1) ⇐⇒ (3) of [23,Theorem 3.1] via results of [3] and [27].…”

“…(Yamashita's proof uses the technology of product systems and Cuntz-Pimsner algebras.) We also use the results of [3] to characterise simplicity of C * (Λ) in terms of the structure of Λ, and to establish a condition under which C * (Λ) is purely infinite.…”

“…As in [3], we say that a topological groupoid G is topologically principal if the set u ∈ G (0) : G u u = {u} of units with trivial isotropy is dense in G (0) , and we say that G is minimal if the only nonempty open invariant subset of G (0) is G (0) itself. Proof.…”

“…So the result follows from Lemma 5.5. Lemma 3.3 of [3] implies that a second-countable locally compact Hausdorff groupoid G is topologically principal if and only if it satisfies the apparently weaker condition (genuinely weaker in the absence of the assumption that G is second countable) that the interior of the isotropy subgroupoid u∈G (0) G u u of G is precisely G (0) . Since this condition should be easier to check, we describe what it says for a topological k-graph: it is a topological analogue of the condition called "no local periodicity" in [20].…”