14:15
The subject of distributional symmetries and invarianceprinciples yields deep results on the structure of the underlying randomobjects. So it is of general interest to investigate if such an approach turnsout to be also fruitful in the quantum world. My talk will report recentprogress in the transfer of de Finetti's pioneering work to noncommutativeprobability. More precisely, an infinite sequence of random variables isexchangeable if its distribution is invariant under finite permutations. The deFinetti theorem characterizes such sequences as conditionally i.i.d. Recentlywe have proven a noncommutative analogue of this celebrated theorem. We willdiscuss the new symmetries `braidability'
and `quantum exchangeability' emerging from our approach.In particular, this brings our approach in close contact with Jones' subfactortheory and Voiculescu's free probability. Finally we will address that ourmethods give a new proof of Thoma's theorem on the general form of charactersof the infinite symmetric group. Quite surprisingly, Thoma's theorem turns outto be the spectral analysis of the tail algebra coming from a certainexchangeable sequence of transpositions. This is in part joint work with RolfGohm and Roland Speicher.
REFERENCES:
[1] C. Koestler. A noncommutative extended de Finettitheorem 258 (2010) 1073-1120.
[2] R. Gohm & C. Kostler. Noncommutativeindependence from the braid group $\mathbb{B}_\infty$. Commun. Math. Phys.289(2) (2009), 435-482.
[3] C. Koestler & R. Speicher. A noncommutative deFinetti theorem:
Invariance under quantum permutations is equivalent tofreeness with amalgamation. Commun. Math. Phys. 291(2) (2009), 473-490.
[4] R. Gohm & C. Koestler: An application ofexchangeability to the symmetric group $\mathbb{S}_\infty$. Preprint.