Noncommutative Independence from the Braid Group B∞

Rolf Gohm, Claus Michael Köstler

Allbwn ymchwil: Cyfraniad at gyfnodolynErthygladolygiad gan gymheiriaid

107 Wedi eu Llwytho i Lawr (Pure)


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.
Iaith wreiddiolSaesneg
Tudalennau (o-i)435-482
Nifer y tudalennau48
CyfnodolynCommunications in Mathematical Physics
Rhif cyhoeddi2
Dyddiad ar-lein cynnar04 Chwef 2009
Dynodwyr Gwrthrych Digidol (DOIs)
StatwsCyhoeddwyd - 01 Gorff 2009

Ôl bys

Gweld gwybodaeth am bynciau ymchwil 'Noncommutative Independence from the Braid Group B∞'. Gyda’i gilydd, maen nhw’n ffurfio ôl bys unigryw.

Dyfynnu hyn