## 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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