A noncommutative central limit theorem for CCR-algebras. (English) Zbl 0571.46043
Let \(\Delta\) and \(\Delta\) ’ be CCR-algebras over nondegenerate symplectic spaces. The central limit theorem tells about the limit of \(T^ n\circ \Phi_ n\) (with an appropriate cutting off) where \(T: \Delta\) \(\to \Delta '\) is a completely positive even unit preserving mapping and \(\Phi_ n:\Delta \to \Delta^ n=\otimes^{n}_{1}\Delta\) is a specified homomorphism. Quasi-free completely positive mappings turn out to be the noncommutative analogue of gaussian distributions since they appear in the central limit. If T is a state then \(T^ n\circ \Phi_ n\) converges pointwise to an even quasi-free state. The construction of the central limit goes as in the case of the CAR-algebra but in the CCR different technical problems arise.
Reviewer: D.Petz
MSC:
46L51 | Noncommutative measure and integration |
46L53 | Noncommutative probability and statistics |
46L54 | Free probability and free operator algebras |
60F05 | Central limit and other weak theorems |
Keywords:
even mappings; conditional expectation; CCR-algebras over nondegenerate symplectic spaces; central limit theorem; completely positive even unit preserving mapping; Quasi-free completely positive mappings; noncommutative analogue of gaussian distributionsReferences:
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