Approximation properties for free orthogonal and free unitary quantum groups

By Michael Brannan. 2011

83 citations recorded by the source corpus.

Abstract In this paper, we study the structure of the reduced C*-algebras and von Neumann algebras associated to the free orthogonal and free unitary quantum groups. We show that the reduced von Neumann algebras of these quantum groups always have the Haagerup approximation property. Combining this result with a Haagerup-type inequality due to Vergnioux, we also show that the reduced C*-algebras always have the metric approximation property.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f906cb091cf7fce2e33c717ff8d440cb859fa66e

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Citations in the corpus

The Haagerup property for locally compact quantum groups (2013, methodology): “The papers [14, 15] study the Haagerup approximation property for L(1).”
Examples of factors which have no Cartan subalgebras (2012, result): “Recently combined with an approximation property result of Brannan [5], Freslon proved that Λc.”
Examples of weakly amenable discrete quantum groups (2012, background): “Brannan in [8] to prove the Haagerup property and the metric approximation property.”
Reduced operator algebras of trace-preserving quantum automorphism groups (2012, methodology): “To check this last fact, note that 0 < Πk(t) Πk(dimB) = S2k( √ t) S2k( √ dimB) , and an easy exercise (see for example [14], Proposition 4.”
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].”
Free Probability and Random Matrices (2014, background): “So we are, for example, not addressing such exciting topics as free stochastic and Malliavin calculus [40, 108, 114], or the rectangular version of free probability [28], or the strong version of asymptotic freeness [48, 58, 89], or free monotone transport [83], or the relation with representation t”
CCAP for Universal Discrete Quantum Groups (2013, methodology): “The following result is due to Brannan but did not appear in the litterature.”