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Completely positive maps on Coxeter groups, deformed commutation relations, and operator spaces. (English) Zbl 0819.20043
Let $$(q_{ij})$$ be a self-adjoint $$n \times n$$ complex matrix, with $$| q_{ij}| \leq 1$$ for all $$i$$, $$j$$. The purpose of this paper is to prove the existence of a Fock representation of the $$q$$-deformed commutation relations. This amounts to proving the existence of a Hilbert space $$H$$, with a distinguished unit vector $$\Omega$$ and operators $$(d_ i)^ n_{i = 1}$$ on $$H$$ fulfilling: $$d_ i \Omega = 0$$ for all $$i$$, $$d_ i d_ j^* - q_{ij} d^*_ j d_ i = \delta_{ij} 1$$ for all $$i$$, $$j$$. The cases of $$q_{ij} \equiv \pm \delta_{ij}$$ correspond to the usual CCR and CAR and are well known. When $$q_{ij} = q \delta_{ij}$$ (with $$| q| \leq 1$$) one gets a model which has been studied recently by Fivel, Greenberg, Zagier, and the authors. In particular, they have shown that the existence of a Fock representation follows from the fact that the function $$\pi \to q^{i(\pi)}$$, where $$i(\pi)$$ is the inversion number of a permutation $$\pi$$, is positive definite on the symmetric group.
In this paper, they extend this analysis by considering some self-adjoint operators $$T_ i$$, of norm $$\leq 1$$, satisfying the braid relations, and proving that the quasi-multiplicative extension of $$\varphi(\pi_ i) = T_ i$$ on $$S_ n$$ is completely positive. (Here $$S_ n$$ is the symmetric group on $$n$$ objects and the $$\pi_ i$$ are its Coxeter generators). In fact they prove a more general version, valid for more general Coxeter groups. Finally, they give an operator space characterization of the span of the operators $$(d_ i)^ n_{i = 1}$$.
Reviewer: Ph.Biane (Paris)

##### MSC:
 20F55 Reflection and Coxeter groups (group-theoretic aspects) 81S05 Commutation relations and statistics as related to quantum mechanics (general) 43A35 Positive definite functions on groups, semigroups, etc. 46L51 Noncommutative measure and integration 46L53 Noncommutative probability and statistics 46L54 Free probability and free operator algebras 20C30 Representations of finite symmetric groups
##### MathOverflow Questions:
Completely positive maps on Coxeter groups - the general case
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