Abstract
AbstractConsider the free orthogonal quantum groups $$O_N^+(F)$$
O
N
+
(
F
)
and free unitary quantum groups $$U_N^+(F)$$
U
N
+
(
F
)
with $$N \ge 3$$
N
≥
3
. In the case $$F = \text {id}_N$$
F
=
id
N
it was proved both by Isono and Fima-Vergnioux that the associated finite von Neumann algebra $$L_\infty (O_N^+)$$
L
∞
(
O
N
+
)
is strongly solid. Moreover, Isono obtains strong solidity also for $$L_\infty (U_N^+)$$
L
∞
(
U
N
+
)
. In this paper we prove for general $$F \in GL_N(\mathbb {C})$$
F
∈
G
L
N
(
C
)
that the von Neumann algebras $$L_\infty (O_N^+(F))$$
L
∞
(
O
N
+
(
F
)
)
and $$L_\infty (U_N^+(F))$$
L
∞
(
U
N
+
(
F
)
)
are strongly solid. A crucial part in our proof is the study of coarse properties of gradient bimodules associated with Dirichlet forms on these algebras and constructions of derivations due to Cipriani–Sauvageot.
Funder
Delft University of Technology
Publisher
Springer Science and Business Media LLC
Reference69 articles.
1. Anantharaman-Delaroche, C.: Amenable correspondences and approximation properties for von Neumann algebras. Pac. J. Math. 171(2), 309–341 (1995)
2. Avitzour, D.: Free products of C$$^\ast $$-algebras. Trans. Am. Math. Soc. 271(2), 423–435 (1982)
3. Avsec, S.: Strong Solidity of the q-Gaussian Algebras for all $$-1 < q < 1$$, arXiv:1110.4918
4. Banica, T.: Le groupe quantique compact libre $$U(n)$$. Commun. Math. Phys. 190(1), 143–172 (1997)
5. Bédos, E., Tuset, L.: Amenability and co-amenability for locally compact quantum groups. Int. J. Math. 14(8), 865–884 (2003)
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