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
ℤ
2
×
{
e
}
≤
(
ℤ
2
×
ℤ
2
)
⋊
ℤ
2
, where the action of
1
¯
∈
ℤ
2
flips the two copies of
ℤ
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
)
≅
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
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
∈
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.
∎
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
≀
ℤ
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
↷
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
.
∎
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