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We prove a limit theorem concerning the sequence of star-generators of [Formula: see text], where the expectation functional is provided by a character of [Formula: see text] with weights [Formula: see text] in the Thoma classification. The limit law turns out to be the law of a “traceless CCR-GUE” matrix, an analogue of the traceless GUE where the off-diagonal entries [Formula: see text] satisfy the commutation relation [Formula: see text]. The special case [Formula: see text] yields the law of a bona fide traceless GUE matrix, and we retrieve a result of Köstler and Nica from 2021, which in turn extended a result of Biane from 1995.

It is well known that, on a purely algebraic level, a simplified version of the central limit theorem (CLT) can be proved in the framework of a non-commutative probability space, under the hypotheses that the sequence of non-commutative random variables we consider is exchangeable and obeys a certain vanishing condition of some of its joint moments. In this approach (which covers versions for both the classical CLT and the CLT of free probability), the determination of the resulting limit law has to be addressed on a case-by-case basis. In this paper we discuss an instance of the above theorem that takes place in the framework of the group algebra $${{\mathbb {C}}}[ S_{\infty } ]$$ of the infinite symmetric group: The exchangeable sequence is provided by the star-generators of $$S_{\infty }$$ , and the expectation functional used on $${{\mathbb {C}}}[ S_{\infty } ]$$ depends in a natural way on a parameter $$d \in {{\mathbb {N}}}$$ . We identify precisely the limit distribution $$\mu _d$$ for this special instance of CLT, via a connection that $$\mu _d$$ turns out to have with the average empirical eigenvalue distribution of a random $$d \times d$$ GUE matrix. Moreover, we put into evidence a multivariate version of this result which follows from the observation that, on the level of calculations with pair-partitions, the (non-centered) star-generators are related to a (centered) exchangeable sequence of GUE matrices with independent entries.

We introduce symmetric states and quantum symmetric states on universal unital free product C*-algebras an arbitrary unital C*-algebra A with itself infinitely many times, as a generalization of the notions of exchangeable and quantum exchangeable random variables. We prove existence of conditional expectations onto tail algebras in various settings and we define a natural C*-subalgebra of the tail algebra, called the tail C*-algebra. Extending and building on the proof of the noncommutative de Finetti theorem of Koestler and Speicher, we prove a de Finetti type theorem that characterizes quantum symmetric states in terms of amalgamated free products over the tail C*-algebra, and we provide a convenient description of the set of all quantum symmetric states on the free product C*-algebra in terms of C*-algebras generated by homomorphic images of A and the tail C*-algebra. This description allows a characterization of the extreme quantum symmetric states. Similar results are proved for the subset of tracial quantum symmetric states, though in terms of von Neumann algebras and normal conditional expectations. The central quantum symmetric states are those for which the tail algebra is in the center of the von Neumann algebra, and we show that the central quantum symmetric states form a Choquet simplex whose extreme points are the free product states, while the tracial central quantum symmetric states form a Choquet simplex whose extreme points are the free product traces., 37 pages. Version 3 corrects a mistake in the earlier versions about existence of a normal conditional expectation onto the tail algebra; and the final results are slightly different than before. Version 2 contains some minor additions to Version 1

We show that any countably generated von Neumann algebra with specified normal faithful state can arise as the tail algebra of a quantum exchangeable sequence of noncommutative random variables. We also characterize the cases when the state corresponds to a limit of convex combinations of free products states.

The extended de Finetti theorem characterizes exchangeable infinite random sequences as conditionally i.i.d. and shows that the apparently weaker distributional symmetry of spreadability is equivalent to exchangeability. Our main result is a noncommutative version of this theorem. In contrast to the classical result of Ryll-Nadzewski, exchangeability turns out to be stronger than spreadability for infinite noncommutative random sequences. Out of our investigations emerges noncommutative conditional independence in terms of a von Neumann algebraic structure closely related to Popa's notion of commuting squares and K\"ummerer's generalized Bernoulli shifts. Our main result is applicable to classical probability, quantum probability, in particular free probability, braid group representations and Jones subfactors., Comment: 44 pages

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.

We consider the concepts of continuous Bernoulli systems and non-commutative white noises. We address the question of isomorphism of continuous Bernoulli systems and show that for large classes of quantum L{\'e}vy processes one can make quite precise statements about the time behaviour of their moments., Comment: 17 pages

To a semi-cosimplicial object (SCO) in a category we associate a system of partial shifts on the inductive limit. We show how to produce an SCO from an action of the infinite braid monoid $\mathbb{B}^+_\infty$ and provide examples. In categories of (noncommutative) probability spaces SCOs correspond to spreadable sequences of random variables, hence SCOs can be considered as the algebraic structure underlying spreadability., 20 pages, minor changes (1.2, 2.9, 4.3) in (v3), to be published in: Rocky Mountain Journal of Mathematics

This is an introductory paper about our recent merge of a noncommutative de Finetti type result with representations of the innite braid and symmetric group which allows us to derive factorization properties from symmetries. We explain some of the main ideas of this approach and work out a constructive procedure to use in applications. Finally we illustrate the method by applying it to the theory of group characters.

We derive the form of the Belavkin-Kushner-Stratonovich equation describing the filtering of a continuous observed quantum system via non-demolition measurements when the statistics of the input field used for the indirect measurement are in a general coherent state. Dedicated to Robin Hudson on the occasion his 70th birthday.

We show that the classical de Finetti theorem has a canonical noncommutative counterpart if we strengthen `exchangeability' (i.e., invariance of the joint distribution of the random variables under the action of the permutation group) to invariance under the action of the quantum permutation group. More precisely, for an infinite sequence of noncommutative random variables, we prove that invariance of their joint distribution under quantum permutations is equivalent to the fact that the random variables are identically distributed and free with respect to the conditional expectation onto their tail algebra., Comment: 17 pages

Inspired by Lehner's results on exchangeability systems, we define 'weak conditional freeness' and 'conditional freeness' for stationary processes in an operator algebraic framework of noncommutative probability. We show that these two properties are equivalent, and thus the process embeds into a von Neumann algebraic amalgamated free product over the fixed point algebra of the stationary process.