Noncommutative independence from the braid group $B_{\infty }$

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. Köstler). 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 $I{I}_1$-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.

minor changes, added 3.3-3.6, version to be published in Comm.Math.Phys. (47 pages)

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• Random matrices, free probability, planar algebras and subfactors
• Generalised Brownian Motion and Second Quantisation
• Randomly growing braid on three strands and the manta ray
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• A transference method in quantum probability
• A noncommutative extended de Finetti theorem
• Conformal Geometry and Invariants of 3-strand Brownian Braids

Cited by in corpus (9)

• Yang-Baxter representations of the infinite symmetric group
• Noncommutative Independence from Characters of the Infinite Symmetric Group $s_{\infty }$
• A noncommutative extended de Finetti theorem
• De Finetti Theorems for the Unitary Dual Group
• Quantum symmetric states on free product C*-algebras
• On Lehner's `free' noncommutative analogue of De Finetti's theorem
• Yang-Baxter endomorphisms
• De Finetti Theorems for Braided Parafermions
• Markovianity and the Thompson Group $F$