## Q-CANONICAL COMMUTATION RELATIONS AND STABILITY OF THE CUNTZ ALGEBRA

Autor(en): | JORGENSEN, PET SCHMITT, LM WERNER, RF |

Stichwörter: | BOSE STATISTICS; CSTAR-ALGEBRAS; EXAMPLE; FERMI; Mathematics; OSCILLATOR |

Erscheinungsdatum: | 1994 |

Herausgeber: | PACIFIC JOURNAL MATHEMATICS |

Journal: | PACIFIC JOURNAL OF MATHEMATICS |

Volumen: | 165 |

Ausgabe: | 1 |

Startseite: | 131 |

Seitenende: | 151 |

Zusammenfassung: | We consider the q-deformed canonical commutation relations a(i)a(j)* - qa(j)*a(i) = delta(ij)1, i, j = 1, ..., d, where d is an integer, and - 1 < q < 1 . We show the existence of a universal solution of these relations, realized in a C*-algebra E(q) with the property that every other realization of the relations by bounded operators is a homomorphic image of the universal one. For q = 0 this algebra is the Cuntz algebra extended by an ideal isomorphic to the compact operators, also known as the Cuntz-Toeplitz algebra. We show that for a general class of commutation relations of the form a(i)a(j)* = GAMMA(ij)(a1 ,..., a(d)) with GAMMA an invertible matrix the algebra of the universal solution exists and is equal to the Cuntz-Toeplitz algebra. For the particular case of the q-canonical commutation relations this result applies for Absolute value of q < square-root 2 - 1 . Hence for these values E(q) is isomorphic to E0. The example a(i)a(j)* - qa(i)*a(j) = delta(ij)1 is also treated in detail. |

ISSN: | 00308730 |

DOI: | 10.2140/pjm.1994.165.131 |

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