  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.
