### Characterization of amenable groups and the Littlewood functions on free groups

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We define an operator α on ℂ³ ⊗ ℂ³ associated with the quantum group ${U}_{q}\left(2\right)$, which satisfies the Yang-Baxter equation and a cubic equation (α² - 1)(α + q²) = 0. This operator can be extended to a family of operators ${h}_{j}:={I}_{j}\otimes \alpha \otimes {I}_{n-2-j}$ on ${\left(\u2102\xb3\right)}^{\otimes n}$ with 0 ≤ j ≤ n - 2. These operators generate the cubic Hecke algebra ${\mathscr{H}}_{q,n}\left(2\right)$ associated with the quantum group ${U}_{q}\left(2\right)$. The purpose of this note is to present the construction.

We present a generalization of the classical central limit theorem to the case of non-commuting random variables which are bm-independent and indexed by a partially ordered set. As the set of indices I we consider discrete lattices in symmetric positive cones, with the order given by the cones. We show that the limit measures have moments which satisfy recurrences generalizing the recurrence for the Catalan numbers.

We show that the von Neumann algebras generated by an infinite number of t-deformed free gaussian operators are factors of type ${I}_{\infty}$.

A family of transformations on the set of all probability measures on the real line is introduced, which makes it possible to define new examples of convolutions. The associated central limit theorems are studied, and examples of the limit measures, related to the classical, free and boolean convolutions, are shown.

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