The boundary of universal discrete quantum groups, exactness, and factoriality

By S. Vaes, Roland Vergnioux. 2005

153 citations recorded by the source corpus.

We study the C -algebras and von Neumann algebras associated with the universal discrete quantum groups. They give rise to full prime factors and simple exact C -algebras. The main tool in our work is the study of an amenable boundary action, yielding the Akemann-Ostrand property. Finally, this boundary can be identified with the Martin or the Poisson boundary of a quantum random walk.

Subfields: Atomic and Molecular Physics, and Optics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/feefdcbdd850870b2cbf48e00d728d63d44c9b10

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In Atomic and Molecular Physics, and Optics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

CCAP for Universal Discrete Quantum Groups (2013, background): “Regarding approximation properties, Vaes and Vergnioux [37] obtained, from a study of the action of the discrete dual FOF on its Martin boundary, the exactness and the Akemann–Ostrand property of the reduced C∗-algebra C∗ r (FOF ).”
Examples of weakly amenable discrete quantum groups (2012, background): “Vergnioux [27] on exactness and factoriality and those of M.”
Quantum permutation groups: a survey (2006, background): “wing properties: (1) A o(2) corresponds to the quantum group SU−1 2 . (2) A o(n) corresponds to an R-matrix quantization of SU2. (3) A u(n) embeds into the free product C(T)∗A o(n). We refer to [10], [49] for an updated discussion of these results. 6 TEODOR BANICA, JULIEN BICHON, AND BENOˆIT COLLINS”
Approximation properties for free orthogonal and free unitary quantum groups (2011, background): “In [3] and [34], the algebras CðGÞ and LyðGÞ were shown to share many structural properties with the reduced C - and von Neumann algebras of the free groups Fk (k f 2).”
Reduced operator algebras of trace-preserving quantum automorphism groups (2012, background): “We also show that when B is a full matrix algebra, L∞(G) is an index 2 subfactor of the reduced von Neumann algebra of a free orthogonal quantum group and thus solid and prime.”
The Baum–Connes conjecture for free orthogonal quantum groups (2009, result): “ be regarded as an analogue of the Kadison-Kaplansky conjecture. Our results support the point of view that free quantum groups behave like free groups in many respects. By work of Vaes and Vergnioux [32] it is known, for instance, that the reduced C∗-algebra C∗ r (FO(n)) of FO(n) is exact and simpl”
Liberation of orthogonal Lie groups (2008, unknown): “Quantum group, Noncrossing partition.”
Approximation properties and absence of Cartan subalgebra for free Araki-Woods factors (2010, unknown): “Shlyakhtenko moreover showed [41] that every free Araki-Woods factor M = Γ(HR, Ut) ′′ is generalized solid in the sense of [26, 54]: for every diffuse subalgebra A ⊂ M for which there exists a faithful normal conditional expectation E : M → A, the relative commutant A′ ∩ M is amenable.”
INVARIANTS OF THE HALF-LIBERATED ORTHOGONAL GROUP (2009, background): “These led to a number of general developments, including the study of connections with subfactors, noncommutative geometry and free probability [1], [5], [14] and a number of advances in relation with operator algebras [16], [17], [18].”
The Connes embedding property for quantum group von Neumann algebras (2014, background): “These connections occur at the lev l of quantum symmetries and asymptotic freeness results [3, 6, 10, 17, 18], and also at th e operator algebra level [9, 19, 21, 22, 23, 35].”
Unique prime factorization and bicentralizer problem for a class of type III factors (2015, unknown): “Examples of von Neumann algebras satisfying the strong condition (AO) include amenable von Neumann algebras, von Neumann algebras associated with bi-exact discrete groups [BO08], von Neumann algebras associated with free quantum groups [Is13b, VV05] and free Araki-Woods factors [Sh96, Ho07].”
ON THE PARTITION APPROACH TO SCHUR-WEYL DUALITY AND FREE QUANTUM GROUPS (2014, background): “However, the conclusion is known to hold also in most of these cases by [42], [15] and [29].”
Examples of factors which have no Cartan subalgebras (2012, unknown): “Then recall from [34, Theorem 7.1] that L ∞ ( G ) then satisﬁes the following conditions: • the algebra L ∞ ( G ) is a full factor and the Haar state h is almost periodic; • the invariant Sd( L ∞ ( G )) is the subgroup Γ of R ∗ + generated by eigenvalues of Q ⊗ Q − 1 , where Q − 1 := FF ∗ .”