92-52 Jorgensen P.E.T., Schmitt L.M., Werner R.F.
q--Canonical Commutation Relations and Stability of the Cuntz Algebra (56K, TeX) May 6, 92
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Abstract. \let\1\sp% \def\E#1{{\cal E}\1{#1}}% \def\idty{{\bf I}}% We consider the $q$-deformed canonical commutation relations $a_ia_j\1*-q\,a_j\1*a_i= \delta_{ij}\idty$, $i,j=1,\ldots,d$, 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 $a_ia_j\1*=\Gamma_{ij}(a_1,\ldots,a_d)$ 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 $\vert q\vert<\sqrt2\,-1$. Hence for these values $\E q$ is isomorphic to $\E0$. The example $a_ia_j\1*-q\,a_i\1*a_j= \delta_{ij}\idty$ is also treated in detail.

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