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research
Exotic C*-algebras of geometric groups
Authors
Ebrahim Samei
Matthew Wiersma
Publication date
19 September 2018
Publisher
View
on
arXiv
Abstract
We consider a new class of potentially exotic group C*-algebras
C
P
F
p
β
β
(
G
)
C^*_{PF_p^*}(G)
C
P
F
p
β
β
β
β
(
G
)
for a locally compact group
G
G
G
, and its connection with the class of potentially exotic group C*-algebras
C
L
p
β
(
G
)
C^*_{L^p}(G)
C
L
p
β
β
(
G
)
introduced by Brown and Guentner. Surprisingly, these two classes of C*-algebras are intimately related. By exploiting this connection, we show
C
L
p
β
(
G
)
=
C
P
F
p
β
β
(
G
)
C^*_{L^p}(G)=C^*_{PF_p^*}(G)
C
L
p
β
β
(
G
)
=
C
P
F
p
β
β
β
β
(
G
)
for
p
β
(
2
,
β
)
p\in (2,\infty)
p
β
(
2
,
β
)
, and the C*-algebras
C
L
p
β
(
G
)
C^*_{L^p}(G)
C
L
p
β
β
(
G
)
are pairwise distinct for
p
β
(
2
,
β
)
p\in (2,\infty)
p
β
(
2
,
β
)
when
G
G
G
belongs to a large class of nonamenable groups possessing the Haagerup property and either the rapid decay property or Kunze-Stein phenomenon by characterizing the positive definite functions that extend to positive linear functionals of
C
L
p
β
(
G
)
C^*_{L^p}(G)
C
L
p
β
β
(
G
)
and
C
P
F
p
β
β
(
G
)
C^*_{PF_p^*}(G)
C
P
F
p
β
β
β
β
(
G
)
. This greatly generalizes earlier results of Okayasu and the second author on the pairwise distinctness of
C
L
p
β
(
G
)
C^*_{L^p}(G)
C
L
p
β
β
(
G
)
for
2
<
p
<
β
2<p<\infty
2
<
p
<
β
when
G
G
G
is either a noncommutative free group or the group
S
L
(
2
,
R
)
SL(2,\mathbb R)
S
L
(
2
,
R
)
, respectively. As a byproduct of our techniques, we present two applications to the theory of unitary representations of a locally compact group
G
G
G
. Firstly, we give a short proof of the well-known Cowling-Haagerup-Howe Theorem which presents sufficient condition implying the weak containment of a cyclic unitary representation of
G
G
G
in the left regular representation of
G
G
G
. Also we give a near solution to a 1978 conjecture of Cowling. This conjecture of Cowling states if
G
G
G
is a Kunze-Stein group and
Ο
\pi
Ο
is a unitary representation of
G
G
G
with cyclic vector
ΞΎ
\xi
ΞΎ
such that the map
G
β
s
β¦
β¨
Ο
(
s
)
ΞΎ
,
ΞΎ
β©
G\ni s\mapsto \langle \pi(s)\xi,\xi\rangle
G
β
s
β¦
β¨
Ο
(
s
)
ΞΎ
,
ΞΎ
β©
belongs to
L
p
(
G
)
L^p(G)
L
p
(
G
)
for some
2
<
p
<
β
2< p <\infty
2
<
p
<
β
, then
A
Ο
β
L
p
(
G
)
A_\pi\subseteq L^p(G)
A
Ο
β
β
L
p
(
G
)
. We show
B
Ο
β
L
p
+
Ο΅
(
G
)
B_\pi\subseteq L^{p+\epsilon}(G)
B
Ο
β
β
L
p
+
Ο΅
(
G
)
for every
Ο΅
>
0
\epsilon>0
Ο΅
>
0
(recall
A
Ο
β
B
Ο
A_\pi\subseteq B_\pi
A
Ο
β
β
B
Ο
β
)
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oai:arXiv.org:1809.07007
Last time updated on 06/02/2019