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.
Mitchell, L. & Paschke, W. (2010). Simplicity of C*-algebras using unique eigenstates. The Journal of Operator Theory, 64(2), 321-347. https://www.theta.ro/jot/archive/2010-064-002/2010-064-002-004.pdf.