Proof for Theorem G.

Since G ⁢ ( m , n ) ≅ ℤ m * ℤ is a discrete Powers group if | m | = n , it suffices to consider the case | m | ≠ n . Fix such m and n and write G = G ⁢ ( m , n ) . Denote by Δ : G → ℝ > 0 the modular function and by G 0 the kernel of Δ. Denote by t ∈ G the image of the free letter. Then G = G 0 ⋊ 〈 t 〉 . If we show that C red * ⁢ ( G 0 ) is simple and has a unique tracial weight, then work of Archbold and Spielberg [1, Corollary, p. 122] shows that also C red * ⁢ ( G ) ≅ C red * ⁢ ( G 0 ) ⋊ ℤ is simple, since the action Ad t on C red * ⁢ ( G 0 ) scales the Plancherel trace by | n m | ≠ 1 .

Denote by K = 〈 a 〉 ¯ ≤ G the elliptic subgroup of G, and recall that G is an HNN-extension of K by Ad t . Denote by σ t : G → ℕ the associated t-exponent sum, as used for example in [3]. Denote by

the t-height of w ∈ G 0 . It is the absolute value of the maximal t-exponent sum of a terminal piece of w. We put

Then G k is a an open subgroup of G 0 and K k = 〈 a m k ⁢ n k 〉 ¯ is a normal compact open subgroup of G k . Since ⋃ k ∈ ℕ G k = G 0 , it suffices to show that Γ k = G k / K k is C * -simple by Suzuki’s result [6].

Denote by T = BS ⁢ ( m , n ) / 〈 a 〉 = G / K the Bass–Serre tree of G, and write ρ for its root. Write T k = G k ⋅ ρ and observe that T k ≤ T is a subtree. Note that the action of K k on T k is trivial, so that we obtain a well-defined action of Γ k on T k . We observe that for every non-empty open subset O ⊂ ∂ ⁡ T k there are infinitely many hyperbolic elements in G k with pairwise different endpoints, which all lie in O. So by the arguments of [2], the group Γ k is a Powers group and thus C * -simple, if we can exhibit for every finite subset F ⊂ Γ k ∖ { e } a non-empty open subset O ⊂ ∂ ⁡ T k such that F ⁢ O ∩ O = ∅ . To this end, it suffices to show that the set of fix points of every non-trivial element of Γ k is meager in ∂ ⁡ T k . Hyperbolic elements have at most two fixed points, while the action of any non-trivial elliptic element in Γ k is conjugate to one of the elements a , a 2 , … , a m k ⁢ n k - 1 , whose fixed points lie in ∂ ⁡ T k - 1 ⊂ ∂ ⁡ T k . Since the latter subset is meager, we can conclude that G is C * -simple. ∎

Proposition 1.

Let G be a locally compact group acting on a compact space X. Assume that the action of G has some open amenable point stabiliser and is micro-supported. Then G is not C * -simple.

Proof.

We may assume that X is non-trivial, since amenable groups are not C * -simple. Let x ∈ X have an open and amenable point stabiliser. Then G ⁢ ↷ ⁢ ℓ 2 ⁢ ( G ⁢ x ) is unitarily equivalent with the quasi-regular representation on G / G x and thus weakly contained in the left-regular representation. Denote by π : C red * ⁢ ( G ) → ℬ ⁢ ( ℓ 2 ⁢ ( G ⁢ x ) ) the associated * -homomorphism. Since the inclusion C red * ⁢ ( G ) ⊂ M ⁢ ( C red * ⁢ ( G ) ) is essential, it suffices to show that the extension π ~ of π to the multiplier algebra M ⁢ ( C red * ⁢ ( G ) ) is not injective. Let U , V ⊂ X be two disjoint open subsets. Let g , h ∈ G be elements acting non-trivially on X such that g fixes X ∖ U pointwise and h fixes X ∖ V pointwise. Then ( 1 - u g ) ⁢ ( 1 - u h ) ∈ ker ⁡ π ~ . Indeed, calculating with the orthogonal basis ( δ y ) y ∈ G ⁢ x of ℓ 2 ⁢ ( G ⁢ x ) , we find ( 1 - u h ) ⁢ δ y = 0 if y ∈ X ∖ V and ( 1 - u h ) ⁢ δ y = δ y - δ h ⁢ y if y ∈ V . In the latter case, we have y , h ⁢ y ∈ V ⊂ X ∖ U . So that ( 1 - u g ) ⁢ ( 1 - u h ) ⁢ δ y follows for all y ∈ G ⁢ x . ∎