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On the Fock representation of the \(q\)-commutation relations. (English) Zbl 0767.46038

We consider the \(C^*\)-algebra \({\mathcal R}^ q\) generated by the representation of the \(q\)-commutation relations on the twisted Fock space. We construct a canonical unitary \(U(= U(q))\) from the twisted Fock space to the usual Fock space, such that \(U{\mathcal R}^ q U^*\) contains the extended Cuntz algebra \({\mathcal R}^ 0\), for all \(q\in(-1,1)\). We prove the equality \(U{\mathcal R}^ q U^*={\mathcal R}^ 0\) for \(q\) satisfying: \[ q^ 2<1-2| q| +2| q|^ 4-2| q|^ 9+\dots+2(-1)^ k | q|^{k^ 2}+\dots\;. \] {}.

MSC:

46L10 General theory of von Neumann algebras
81S05 Commutation relations and statistics as related to quantum mechanics (general)
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