Noncommutative Independence from the Braid Group B ∞

We introduce 'braidability' as a new symmetry for (infinite) sequences of noncommutative random variables related to representations of the braid group B∞. It provides an extension of exchangeability which is tied to the symmetric group S∞. Our key result is that braidability implies spread-ability and thus conditional independence, according to the noncommutative extended de Finetti theorem [Kös08]. This endows the braid groups Bn 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 endomor-phisms [Goh04] 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∞ and the irreducible subfactor with infinite Jones index in the non-hyperfinite II 1-factor L(B∞) related to it. Our investigations reveal a new presentation of the braid group B∞, the 'square root of free generator presentation' F 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 [GJS07]; and we speculate about braidability as an extension of (amalgamated) freeness on the combinatorial level.