Acta Mathematica Academiae Scientiarum Hungaricae 63. (1994)

1994 / 3. szám - Accardi, L. - Lu, Y. G.: Quantum central limit theorems for weakly dependent maps. II

Acta Math. Hungar. 63 (3) (1994), 249-282. QUANTUM CENTRAL LIMIT THEOREMS FOR WEAKLY DEPENDENT MAPS. II L. ACCARDI and Y. G. LU* (Roma) Introduction In Part I ([21]), three central limit theorems have been stated: the first one (Theorem (1.3)) includes a law of large numbers and is a vanish­ing result; the second one (Theorem (1.4)) is a central limit theorem for processes with discrete parameter and the third result (Theorem (1.5)) is the extension of the second one to the case of continuous parameter. Our central limit theorems are deformations of the usual quantum central limit theorems, considered up to now, in three different ways: i) The factor a{b,b') in the commutation relations (*) jt(b)js(b') = o(t,s,b,b')jt(b')jt(b) + e(t,s,b,b') (b,b' E В С В', В is the set of algebraic generators of В) are not restricted to the values ±1. ii) The factor e(t,s,b,b') in (*) is not required to vanish identically. iii) Independence is replaced by weak dependence. First of all the states that we obtain in the limit are of Gaussian type, in the sense that their odd momenta vanish and the even ones are given by weighted sums of products of pair correlations. However, while the £-factor simply produces a shift in the correlation function of the limiting state (cf. (1.17) of Part I); the a-factor can give rise to more interesting phenomena. In fact, if all the a(b,b') are present, then the final expressions (1.18) or (1.24) in Part I differ from the usual expressions for the even momenta of a Gaussian state only for the presence of a combinatorial factor. However, if some of the <7(6,6') are allowed to vanish, then the sums (1.18) and (1.24) will no longer be over all pair partitions, but only over a subset of them. Now it is well known that the notion of free independence, recently introduced by Voiculescu [16], leads naturally, by means of free central limit theorems to a notion of free Gaussianity characterized, in terms of momenta, precisely by the property that the summation in the expression of even momenta is *On leave of absence from Beijing Normal University.

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