Q-CANONICAL COMMUTATION RELATIONS AND STABILITY OF THE CUNTZ ALGEBRA

Abstract

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.