We consider the q-deformed
canonical commutation relations aiaj∗− qaj∗ai= δij1, 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 ℰq 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 aiaj∗= Γij(a1,…,ad) with Γ 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 |q| <− 1. Hence for these values ℰq
is isomorphic to ℰ0. The example aiaj∗− qai∗aj= δij1 is also treated in
detail.
