Erratum to C*-simplicity of locally compact Powers groups (J. reine angew. Math. 748 (2019), 173–205)

Sven Raum
This erratum corrects the original online version which can be found here: https://doi.org/10.1515/crelle-2016-0026

In [5, Lemma 5.1] we made the following claim. We claimed that if G is a locally compact group and K G is a compact open subgroup, then for all g G we have

p K u g * p K u g p K [ K : K g K g - 1 ] - 2 p K .

As previously reported in [4], Yuhei Suzuki pointed out that this claim is not correct and provided the counterexample 2 × { e } ( 2 × 2 ) 2 , where the action of 1 ¯ 2 flips the two copies of 2 . The mistake in the proof of [5, Lemma 5.1] lies in the second equality of the first displayed equation, which exchanges a square of a sum with the sum of the squares.

It is a priori not clear whether the invalidity of [5, Lemma 5.1] affects any of the main results of [5]. In this erratum, we show that the main example of a non-discrete C * -simple group obtained in Theorem G persists, while Theorem B is not correct in the stated generality.

In [5, Theorem G] we claimed that given 2 | m | n the relative profinite completion G ( m , n ) of the Baumslag–Solitar group BS ( m , n ) is C * -simple. We sketch an alternative proof for this statement, which does avoid the use of [5, Lemma 5.1].

Proof for Theorem G.

Since G ( m , n ) m * 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 > 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 ) 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 the associated t-exponent sum, as used for example in [3]. Denote by

h t ( w ) = max { | σ t ( v ) | w = u v  reduced decomposition }

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

G k = { w G 0 h t ( w ) k } .

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 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. ∎

As explained above, [5, Theorem B] is not correct as stated. A counterexample is provided by the Burger–Mozes group U ( F ) associated with the permutation group 2 S 3 with its imprimitive action on a set with six elements. This counterexample as well as the following criterion for non- C * -simplicity were suggested to us by Pierre-Emmanuel Caprace. We thank him for allowing us to include them into this piece. An action of a group G on a space X is called micro-supported if for all non-empty open subsets U X there is some g G acting non-trivially on X and fixing X U pointwise. Note that the action of any Burger–Mozes group on the boundary of its tree is micro-supported, thanks to Tits’ independence property.

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 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 ) ( 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 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 . ∎

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