Stability of a special class of \(q_{ij}\)-CCR and extensions of irrational rotation algebras (Q2722114)

The author studies the \(C^*\)-algebra \(A_{\{q_i\},\{\lambda_{ij}\}}\) corresponding to the relations \(a_i^*a_i=1+q_ia_ia_i^*\) (\(-1<q_i<1\)), \(a_i^*a_j=\lambda_{ij}a_ja_i^*\) (\(|\lambda_{ij}|=1,\lambda_{ij}=\overline{\lambda_{ji}}\)), \(a_ja_i=\lambda_{ij}a_ia_j\), \(i,j=1,\ldots ,d\). It is shown that \(A_{\{q_i\},\{\lambda_{ij}\}}\) is isomorphic to the \(C^*\)-algebra \(A_{\{0\},\{\lambda_{ij}\}}\) generated by isometries. The latter is nuclear. Its irreducible representations are described. The case \(d=2\) is considered in details.