Simplicity of C*-algebras using unique eigenstates

We consider a one-parameter family of operators that are constructed from a pair of isometries on Hilbert space with orthogonal ranges. For special values of the parameter, the operator plays a role in the representation theory of free groups and in free probability theory. For each parameter value, we identify the irreducible ∗-representations of the pair of isometries in which the operator has an eigenvalue. This yields a new technique for showing that certain C ∗ -algebras, including the C ∗ -algebra generated by the operator, are simple. We establish several other fundamental properties of this C ∗ -algebra and its generator