De Finetti theorems for easy quantum groups

By T. Banica, Stephen J. Curran, Roland Speicher. 2009

71 citations recorded by the source corpus.

We study sequences of noncommutative random variables which are invariant under “quantum transformations” coming from an orthogonal quantum group satisfying the “easiness” condition axiomatized in our previous paper. For 10 easy quantum groups, we obtain de Finetti type theorems characterizing the joint distribution of any infinite quantum invariant sequence. In particular, we give a new and unified proof of the classical results of de Finetti and Freedman for the easy groups Sn, On, which is based on the combinatorial theory of cumulants. We also recover the free de Finetti theorem of Kostler and Speicher, and the characterization of operator-valued free semicircular families due to Curran. We consider also finite sequences, and prove an approximation result in the spirit of Diaconis and Freedman.

Subfields: Geometry and Topology, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/81fb8ae481ca7a942aec4c834464060f5fcbba33

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. Moritz Weber
2. Sven Raum
3. Roland Speicher
4. Michael Brannan
5. Amaury Freslon
6. Yusuke Isono

In Mathematical Physics

1. Moritz Weber
2. Sven Raum
3. Roland Speicher
4. Michael Brannan
5. Amaury Freslon
6. Pierre Tarrago
7. Yusuke Isono

In Statistics and Probability

1. Sven Raum
2. Moritz Weber
3. Roland Speicher
4. Michael Brannan

Citations in the corpus

• Easy quantum groups and quantum subgroups of a semi-direct product quantum group (2013, background): “In order to describe corepresentation categories of compact quantum matrix groups combinatorially, Banica and Speicher introduced the notions of a category of partitions and of easy quantum groups [BS09].”
• On the classification of easy quantum groups (2012, background): “nts at the deep connection between co-actions of (easy) quantum groups and free probability theory, which lead to further work by Banica, Curran and Speicher (see also [Cr09], [Cr10], [Cr11], [CS11], [BCS12]). In the case of classical easy quantum groups (which are in fact groups) Sn ⊆ G ⊆ On, the c”
• Free Probability and Random Matrices (2014, background): “This has triggered quite some investigations on more general quantum symmetries and its relations to de Finetti theorems, see [4, 5, 32, 52].”
• Approximation properties for free orthogonal and free unitary quantum groups (2011, background): “i,j≤ N). Since the introduction of these quantum groups, the series {O+ N}N≥2 and {U+ N}N≥2 have been the object of intensive study from both quantum probabilistic and operator algebraic perspectives [2, 3, 5, 6, 7, 16, 34, 35]. On the probabilistic side, it is becoming increasingly apparent that th”
• 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.”