Unbounded representations of $q$-deformation of Cuntz algebra
  We study a deformation of the Cuntz-Toeplitz $C^*$-algebra determined by the relations $a_i^*a_i=1+q a_ia_i^*, a_i^*a_j=0$. We define well-behaved unbounded *-representations of the *-algebra defined by relations above and classify all such irreducible representations up to unitary equivalence. 
Vasyl Ostrovskyi, Daniil Proskurin, Lyudmila Turowska