Noncommutative Independence from the Braid Group $${mathbb{B}_{infty}}$$

We introduce ‘braidability’ as a new symmetry for infinite sequences of noncommutative random variables related to representations of the braid group ({mathbb{B}_{infty}}) . It provides an extension of exchangeability which is tied to the symmetric group ({mathbb{S}_{infty}}) . Our key result is that braidability implies spreadability and thus conditional independence, according to the noncommutative extended de Finetti theorem [Kös08]. This endows the braid groups ({mathbb{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 [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 ({mathbb{B}_{infty}}) and the irreducible subfactor with infinite Jones index in the non-hyperfinite I I ₁-factor L ({(mathbb{B}_{infty})}) related to it. Our investigations reveal a new presentation of the braid group ({mathbb{B}_{infty}}) , the ‘square root of free generator presentation’ ({mathbb{F}^{1/2}_{infty}}) . 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.