 9252 Jorgensen P.E.T., Schmitt L.M., Werner R.F.
 qCanonical 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
CuntzToeplitz 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 CuntzToeplitz 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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