Noncommutative Independence from the Braid Group B∞

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dc.contributor.author Gohm, Rolf
dc.contributor.author Köstler, Claus Michael
dc.date.accessioned 2010-01-11T18:26:52Z
dc.date.available 2010-01-11T18:26:52Z
dc.date.issued 2009-07-01
dc.identifier.citation Gohm , R & Köstler , C M 2009 , ' Noncommutative Independence from the Braid Group B∞ ' Communications in Mathematical Physics , vol 289 , no. 2 , pp. 435-482 . en
dc.identifier.issn 0010-3616
dc.identifier.other PURE: 143115
dc.identifier.other dspace: 2160/3975
dc.identifier.uri http://hdl.handle.net/2160/3975
dc.identifier.uri http://xxx.lanl.gov/abs/0806.3691 en
dc.description Comm. Math. Phys., vol.289(2) (2009), 435-482 en
dc.description.abstract We introduce `braidability' as a new symmetry for (infinite) sequences of noncommutative random variables related to representations of the braid group $B_\infty$. It provides an extension of exchangeability which is tied to the symmetric group $S_\infty$. Our key result is that braidability implies spreadability and thus conditional independence, according to the noncommutative extended de Finetti theorem (of C. K\'{o}stler). This endows the braid groups $B_n$ with a new intrinsic (quantum) probabilistic interpretation. We underline this interpretation by a braided extension of the Hewitt-Savage Zero-One Law. Furthermore we use the concept of product representations of endomorphisms (of R. Gohm) with respect to certain Galois type towers of fixed point algebras to show that braidability produces triangular towers of commuting squares and noncommutative Bernoulli shifts. As a specific case we study the left regular representation of $B_\infty$ and the irreducible subfactor with infinite Jones index in the non-hyperfinite $II_1$-factor $L(B_\infty)$ related to it. Our investigations reveal a new presentation of the braid group $B_\infty$, the `square root of free generator presentation' $F_\infty^{1/2}$. These new generators give rise to braidability while the squares of them yield a free family. Hence our results provide another facet of the strong connection between subfactors and free probability theory and we speculate about braidability as an extension of (amalgamated) freeness on the combinatorial level. en
dc.format.extent 48 en
dc.language.iso eng
dc.relation.ispartof Communications in Mathematical Physics en
dc.title Noncommutative Independence from the Braid Group B∞ en
dc.type Text en
dc.type.publicationtype Article (Journal) en
dc.identifier.doi http://dx.doi.org/10.1007/s00220-008-0716-x
dc.contributor.institution Mathematics and Physics en
dc.contributor.institution Institute of Mathematics & Physics (ADT) en
dc.description.status Peer reviewed en


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