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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
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References:
[1] Blecher, D., Paulsen, V.: Tensor products of operator spaces. J. Funct. Anal.99, 262–292 (1991) · Zbl 0786.46056
[2] Bo\.zejko, M.: Positive definite kernels, length functions on groups and a non commutative von Neumann inequality. Studia Math.95, 107–118 (1989) · Zbl 0714.43007
[3] Bo\.zejko, M.: In preparation
[4] Bo\.zejko, M.: Positive and negative definite kernels on discrete groups. Springer Lect. Notes (to appear)
[5] Bo\.zejko, M., Januszkiewicz, T., Spatzier, R.J.: Infinite Coxeter groups do not have Kazhdan’s property. J. Operator Theory19, 63–68 (1988) · Zbl 0662.20040
[6] Bo\.zejko, M., Speicher, R.: An Example of a Generalized Brownian Motion. Commun. Math. Phys.137, 519–531 (1991) · Zbl 0722.60033
[7] Bo\.zejko, M., Speicher, R.: An example of a generalized Brownian motion II. In: Quantum Probability and Related Topics VII (ed. L. Accardi, pp. 67–77), Singapore: World Scientific 1992 · Zbl 0797.60066
[8] Bo\.zejko, M., Speicher, R.: Interpolations between bosonic and fermionic relations given by generalized Brownian motions. Preprint SFB 123-691, Heidelberg 1992
[9] Bourbaki, N.: Groupes et algebres de Lie, Chap. 4, 5, 6. Paris: Hermann 1968 · Zbl 0186.33001
[10] Carter, R.W.: Simple groups of Lie type. New York: John Wiley & Sons 1972 · Zbl 0248.20015
[11] de la Harpe, P.: Groupes de Coxeter non affines. Expos. Math.5, 91–95 (1987) · Zbl 0605.20049
[12] Effros, E., Ruan, Z.J.: On matricially normed spaces. Pac. J. Math.132, 243–264 (1988) · Zbl 0686.46012
[13] Effros, E., Ruan, Z.J.: A new approach to operator spaces. Can. Math. Bull.34, 329–337 (1991) · Zbl 0769.46037
[14] Evans, D.E.: OnO n. Publ. RIMS16, 915–927 (1980) · Zbl 0461.46042
[15] Fivel, D.: Interpolation between Fermi and Bose statistics using generalized commutators. Phys. Rev. Lett.65, 3361–3364 (1990); Erratum: Phys. Rev. Lett.69, 2020 (1992) · Zbl 1050.81567
[16] Greenberg, O.W.: Particles with small violations of Fermi or Bose statistics. Phys. Rev. D43, 4111–4120 (1991)
[17] Haagerup, U., Pisier, G.: Bounded linear operators betweenC *-algebras. Preprint 1993. Duke Math. J. (to appear) · Zbl 0803.46064
[18] Jimbo, M.: Yang-Baxter equation in integrable systems. Adv. Ser. Math. Phy. 10, Singapore: World Scientific 1989 · Zbl 0697.35131
[19] Jørgensen, P.E.T., Schmitt, L.M., Werner, R.F.:q-Canonical Commutation Relations and Stability of the Cuntz Algebra. Preprint 1992. Pac. J. Math. (to appear) · Zbl 0808.46094
[20] Jørgensen, P.E.T., Schmitt, L.M., Werner, R.F.: Positive representations of general Wick ordering commutation relations. Preprint 1993 · Zbl 0864.46047
[21] Manin, Y.I.: Topics in Noncommutative Geometry. Princeton: Princeton University Press 1991 · Zbl 0724.17007
[22] Paulsen, V.: Completely bounded maps and dilations. Pitna Research Notes 146, New York: John Wiley & Sons 1986 · Zbl 0614.47006
[23] Pisier, G.: Multipliers and lacunary sets in non-amenable groups. Preprint 1992. Am. J. Math. (to appear) · Zbl 0779.43002
[24] Pisier, G.: The Operator Hilbert Space OH, Complex Interpolation and Tensor Norms. Preprint 1993. Mem. AMS (to appear) · Zbl 0803.47039
[25] Solomon, L.: The orders of the finite Chevalley groups. J. Algebra3, 376–393 (1966) · Zbl 0151.02003
[26] Speicher, R.: A New Example of ”Independence” and ”White Noise”. Probab. Th. Rel. Fields84, 141–159 (1990) · Zbl 0671.60109
[27] Speicher, R.: Generalized Statistics of Macroscopic Fields. Lett. Math. Phys.27, 97–104 (1993) · Zbl 0850.46022
[28] Voiculescu, D.: Symmetries of some reduced free productC *-algebras. In: Operator Algebras and their Connection with Topology and Ergodic Theory (LNM 1132, pp. 556–588), Heidelberg: Springer 1985
[29] Voiculescu, D., Dykema, K., Nica, A.: Free Random Variables. AMS 1992 · Zbl 0795.46049
[30] Wassermann, S.: InjectiveW *-algebras. Math. Proc. Camb. Phil. Soc.82, 39–47 (1977) · Zbl 0372.46065
[31] Wenzl, H.: Representations of braid groups and the quantum Yang-Baxter equation. Pac. J. Math.145, 153–180 (1990) · Zbl 0735.57004
[32] Zagier, D.: Realizability of a Model in Infinite Statistics. Comm. Math. Phys.147, 199–210 (1992) · Zbl 0789.47042
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